  This is a brief survey of recent results by the authors devoted to one of the
most important operators of integral geometry. Basic facts about the analytic
family of cosine transforms on the unit sphere and the corresponding Funk
transform are extended to the "higher-rank" case for functions on Stiefel and
Grassmann manifolds. The main topics are the analytic continuation and the
structure of polar sets, the connection with the Fourier transform on the space
of rectangular matrices, inversion formulas and spectral analysis, and the
group-theoretic realization as an intertwining operator between generalized
principal series representations of SL(n, R).
