1 Introduction

The magnetically dominated plasma of the solar corona is an elastic and compressible medium which can support propagation of various types of waves. For a large class of waves with wavelengths and periods large compared with the ion Larmor radius ( $< 1 m$ for almost all combinations of coronal parameters) and the gyroperiod ( $< 10-4 s$ ), respectively, the waves can be described using magnetohydrodynamics (MHD). These waves perturb macro-parameters of the coronal plasma, such as density, temperature, bulk velocity and the frozen-in magnetic field. For wavelengths comparable with the characteristic sizes of coronal plasma structures (e.g., loop major and minor radii, widths of plumes and helmet structures, size of active regions, and so on), the typical periods are in the range from a few seconds to several minutes. This range is well covered by temporal resolution of presently available ground-based and spaceborne observational tools, allowing for the observational detection of the waves. However, it is very important to understand that in normal circumstances both spatial and temporal resolution are necessary ingredients of successful detection of the waves: e.g., the pixel size must be much smaller than the wavelength (taking into account the projection effect!) and the cadence time must be shorter than the period. Only in certain exceptional cases, e.g., when the wave passes through a bright object with the geometrical size smaller than the wavelength, the wave can be detected with poorer spatial resolution.

The launch of SOHO and TRACE spacecrafts led a revolutionary breakthrough in the observational study of coronal wave activity, including the discoveries of EIT (or coronal Moreton) waves (Thompson et al., 1998 ), compressible waves in polar plumes (Ofman et al., 1997 ; DeForest and Gurman, 1998 ; Ofman et al., 1999 ) and in coronal loops (Berghmans and Clette, 1999 ; De Moortel et al., 2000 ; Robbrecht et al., 2001 ), flare-generated global kink oscillations of loops (Aschwanden et al., 1999 , 2002 ; Nakariakov et al., 1999 ; Schrijver et al., 2002 ), and longitudinal standing oscillations within loops (Kliem et al., 2002 ; Wang et al., 2002 ). Also, ground-based radio observations have long revealed the presence of periodic and quasi-periodic phenomena in the corona (Aschwanden, 1987 ). Moreover, very recently ground-based optical observations of eclipses have revealed the presence of rapid oscillations in coronal loops (Williams et al., 2001 , 2002 ). Extensive observational reviews of this diversity of coronal oscillations are given in Aschwanden (2003 ); Nakariakov (2003 ); Aschwanden (2004 ). The current state of the theoretical studies of MHD wave propagation in the solar atmosphere is discussed in Roberts (2000 , 2002 ); Goossens et al. (2002b ); Roberts and Nakariakov (2003 ); Roberts (2004 ).

MHD waves in the corona have been intensively investigated for more than two decades, primarily in the context of the enigmatic problems of coronal heating and acceleration of the fast solar wind. Discussion of those issues is out of scopes of this review and may be found in Walsh and Ireland (2003 ); Cranmer (2004 ); Ofman (2004 ). It is also believed that the coronal MHD waves play an important role in solar-terrestrial connections. The investigation of the coronal waves is now an essential part of solar physics, space physics, geophysics and astrophysics.

The aim of this review is to reflect the current trends in the observational study of coronal wave and oscillatory phenomena, their phenomenology, their interpretation in terms of MHD wave theory, latest achievements in the theoretical modelling of interaction of MHD waves with inhomogeneous plasmas and its relevance to the coronal waves, and recent progress in MHD coronal seismology.

1.1 The method of MHD coronal seismology