  Modern observatories have revealed the ubiquitous presence of
magnetohydrodynamic waves in the solar corona. The propagating waves (in
contrast to the standing waves) are usually originated in the lower solar
atmosphere which makes them particularly relevant for coronal heating.
Furthermore, open coronal structures are believed to be the source regions of
solar wind, therefore, the detection of MHD waves in these structures is also
pertinent to the acceleration of solar wind. Besides, the advanced capabilities
of the current generation telescopes have allowed us to extract important
coronal properties through MHD seismology. The recent progress made in the
detection, origin, and damping of both slow mangetoacoustic waves and
Alfv\'{e}nic waves is presented in this review article especially in the
context of open coronal structures. Where appropriate, we give an overview on
associated theoretical modelling studies. A few of the important seismological
applications of these waves are discussed. The possible role of Aflv\'{e}nic
waves in the acceleration of solar wind is also touched upon.
