This paper provides a comprehensive review of recent theoretical investigations concerning magnetohydrodynamic (MHD) waves in partially ionized solar plasma. First, we examine the properties of linear MHD waves in a uniform partially ionized plasma and discuss the relevant effects arising from partial ionization. Subsequently, we delve into MHD wave studies in the more intricate settings of the lower solar atmosphere and solar prominences. These investigations involve topics such as MHD waves in magnetic flux tubes, wave excitation, linear and nonlinear mode coupling and wave heating. We outline new challenges that future research should tackle. This article is part of the theme issue 'Partially ionized plasma of the solar atmosphere: recent advances and future pathways'.
Keywords: magnetohydrodynamic waves; partially ionized plasma; solar atmosphere.