Fundamental aspects of pulsation theory are brie y discussed such as the solution of the hydrodynamic equations by means of perturbation theory and the important role of opacity. The concept of pressure and gravity modes is introduced. A discussion of regularities in frequency and period spacing in the asymptotic limit leads to the idea of using the large and small separations for estimating the mean density and age of the star. The Petersen diagram is often used for radial modes, but results can be a ected by metallicity and rotation. The e ects of rotation on the pulsation frequencies profound: even slow rotation can destroy the regularities in frequency and period spacing predicted by the asymptotic approximation. Nonperturbative methods of solution for models of rapidly rotating stars o er promising prospects. The excitation mechanisms at work in di erent classes of stars are discussed. The role of convection in driving and damping of pulsations is still poorly understood, in spite of recent advances. Solution of the full nonlinear pulsation equations is the ultimate goal, but at present this is still con ned to radial pulsation. Nonlinear e ects which determine the mode amplitude are brie y discussed. Much e ort has been devoted to develop methods which allow mode identi cation using multicolour photometry or high-resolution spectroscopy. The advantages and drawbacks of each method discussed. In the conclusion, the topics which require further study are highlighted and brieefly discussed.

Although periodic light variability is rare among O-type stars, periodic line pro le variations are more common. The modes in these stars are probably all of high degree and the visibility may be a ected by obscuration. The observed frequencies in β Cep stars cover a wider range than predicted. This is a major unsolved problem which may point to problems in the current opacities. Observations of these stars in low-metallicity environments also present a challenge. Asteroseismology of these stars not been very successful due to the e ect of rotation and the fact that some observed modes are stable in the models. For the SPB stars, the decrease in amplitude with rotation is unexplained. The Be stars are clearly multiperiodic variables, but most of the periods are unresolved. Recent space observations suggest that some of the variability is probably due to rotational modulation. An alternative view is that the light and line pro le variations are due to obscuring clouds locked into co-rotation by a magnetic eld. The fact that the rotational velocities of early type Be stars are in general well below the critical velocity suggest that pulsation is not the trigger for mass loss. Recent space observations suggest that there may exist pulsating stars in the region between the SPB and δ Scuti instability strips. Models of pulsating supergiants are located in a high-luminosity extension of the β Cep and SPB instability strips, but observations do not support these calculations. Models of Wolf-Rayet stars invariably predict instability due to the strange mode mechanism, but like the O-type stars, observations of periodic variations are almost entirely absent.

The use of time-dependent convection has partially resolved the problem of the red edge of the δ Scuti instability strip. Both the κ mechanism and convective blocking are active driving agents in these stars. Recent space observations have destroyed the separation between δ Sct and γ Dor stars in the HR diagram. It seems that a cool δ Sct star is distinguished from a γ Dor star only in the relative amplitudes of the low and high frequencies and that hybrid behaviour is normal in most of these stars. In some δ Sct stars there are a large number of observed frequencies and in a few of these stars the spherical harmonic degree of many of these modes known. In spite of this, asteroseismology of these stars has not been successful. High-amplitude δ Sct stars offer interesting viewof the transition between δ Sct and Cepheid pulsations and provide valuable insights on the mode selection mechanism. The nature of the pulsations in peculiar stars such as λ Boo, Am and Ap stars still need to be understood. Recent models of Am stars taking disusion into account have led to a better understanding of the location of these stars in the HR diagram. However the detection of several pulsating Ap stars from recent space observations may pose a challenge. The number of pulsating pre-main sequence stars has increased greatly in recent years, but modelling of these stars has been slow. Attempts to detect predicted pulsation in brown dwarfs are also discussed.

Rapidly pulsating Ap stars present the widest range of physical phenomena in any pulsating star. It seems probable that most of the driving is due to the κ mechanism in the H ionization zone in which convection is suppressed by the magnetic eld. Recent attempts to model some stars show that driving by this mechanism is sometimes insu cient and that another unknown mechanism is also at work. This is also indicated by the fact that the predicted instability strip for the κ mechanism is much hotter than observed. Much progress has occurred in the last decade and we now have a good understanding of the cyclic frequency jumps that occur in models as the magnetic eld is changed. Coupling of the pulsation with the magnetic eld induces a slow wave which is thought to damp δ Scuti pulsations in these stars. However, recent space observations have shown that in at least one roAp star a low frequency mode is also present, presenting a challenge to this idea. The eigenfunctions are far from simple spherical harmonics, considerably complicating mode identi cation. Observations indicate the presence of pulsational nodes in the atmosphere and variations in amplitude and phase shifts between di erent atomic species. This appears to be the result of chemical strati cation and running waves. The line pro le variations are axisymmetric except in the uppermost atmospheric layers. The non-axisymmetric variations are not understood, but may be a result of turbulence in the atmosphere or shock waves.

Nonlinear Cepheid models have recently allowed stellar parameters to be estimated from the shape of the light curve. Double-mode Cepheids o er even more constraints, allowing the metallicity to be derived as well. They have been used as metallicity probes in various stellar systems. The problem of what sustains doublemode pulsation appears to be controversial and is still unresolved. In the LMC there are more double-mode Cepheids pulsating in the rst and second overtone than in the fundamental and rst overtone. Another puzzle concerns the rate of period change in triple-mode Cepheids. Ultra low-amplitude Cepheids seem to follow a di erent period - luminosity (PL) law from classical Cepheids and present an interesting challenge, as does the discovery of nonradial pulsations in a few LMC Cepheids. The projection factor to be used in a Baade-Wesselink analysis is discussed. There is a small dependence of the Cepheid PL relation on metallicity and it is probably slightly nonlinear. The di erences in the PL relations between classical Cepheids, the anomalous Cepheids and between the BL Her, W Vir and RV Tau stars that comprise the Population II Cepheids are discussed. Recent studies indicate that anomalous Cepheids may have very low metal abundances. Nonlinear modelling of BL Her stars has been quite successful in explaining the shapes of the light curves. A recent modelling survey of RR Lyrae stars has greatly improved our understanding of the Oosterho groups. The physical parameters of RR Lyrae stars can be determined by comparing the shape of the light curve with model calculations. The origin of the Blazhko e ect still remains one of the greatest unsolved puzzles.

The theory of stochastically driven oscillations usually predicts amplitudes which are smaller than actually observed. This could be due to the adoption of an incorrect distribution of the eddy time correlation. Predictions of the amplitudes, frequencies and mode lifetimes are usually made on the basis of scaling relations based on the Sun and are only crude approximations. The modes have short lifetimes and relatively high frequencies, which means that the asymptotic approximation is generally a good description. Since rotation is low, these are ideal stars for asteroseismology. About 40 stars of this type have been identi ed from ground-based high precision radial velocities. In most cases only a hump of excess power is seen in the frequency spectrum, but the large and small separations can sometimes be determined. In some cases there is no agreement in frequencies between di erent observation sets. High precision photometric observations from space are rapidly changing the situation, but problems exist even here. Red giants have the largest solar-like oscillation amplitudes, but the frequencies may not be in the asymptotic region. The modes are not well resolved due to the short lifetimes. Future progress for solar-like oscillations is likely to rest almost entirely on space observations. Recent space observations have conclusively demonstrated that the the pulsations in red giants are indeed global eigenfrequencies.

In the period - luminosity (PL) diagram of pulsating red giants in the LMC, five sequences can be identified. Recent observations have refined this initial finding and many more sequences have been discovered. The two main sequences comprise Miras and semiregulars which are stars in the fundamental and first-overtone radial pulsation modes. Carbon-rich and oxygen-rich Miras form distinct sub-sequences as do C-rich and O-rich semiregulars. At least four further sequences roughly parallel to the Mira and semiregular sequences have been identified which may correspond to higher overtone radial modes. The stars which populate these sequences are the OGLE Small Amplitude Red Giants (OSARGs). The OSARG sequences themselves are resolved into further subsequences of asymptotic giant branch stars and stars below the tip of the red giant branch. It is likely that nonradial modes are present in OSARGs as the Petersen diagram for these stars show some period ratios close to unity. In addition to these complex sequences of pulsating stars, there are two more sequences in the PL diagram. One sequence can be shown to consist of contact binaries. The nature of stars in the other sequence is a mystery. These stars have long secondary periods which do not seem to be due to pulsation. Although it is certain that convection is responsible for driving pulsation in these stars, the precise driving mechanism is not understood. It is possible that in the low-amplitude stars pulsation may be driven stochastically.

ZZ Ceti, V777 Her and DQV stars are white dwarfs with pulsations excited by convective driving. The highly distorted light curves in some of these stars provide information about the convective zone. In V777 Her itself, rotationally-split multiplets allow mode identi cation and a successful asteroseismological solution. The frequency spacings of the multiplets vary in a way which is not understood. A remarkable event occurred some years ago when V777 Her suddenly changed its pulsation characteristics in a very short time. Pulsations in DQ stars are probably caused by convective driving in the ionization zones of HeII or C (or both). Two recent suggestions for the so-called DB gap are discussed. Models indicate that the nuclear ε mechanism may be e ective in certain regions of the HR diagram, but so far no variables have been found in the predicted regions. The rates of period change in the pre-white dwarf GW Vir stars are puzzling. GW Vir itself has a very large number of frequencies and rotationally split multiplets which has led to a successful asteroseismological study. Di usion increases the abundance of Fe-group elements in the driving region of subdwarf B stars, leading to pulsations driven by the κ mechanism. While models can successfully reproduce the pulsation periods in the short-period V361 Hya group of sdB stars, this is not the case for the long-period V1093 Her group. The location of the instability strip and periods in these stars present a challenge. A number of asteroseismological solutions exist for V361 Hya stars by direct matching of observed and calculated periods. Recently a sdO star was found to be pulsating. Pulsations in He-rich subdwarfs and in extreme helium B stars are also discussed.