A regular subriemannian manifold $M$ carries a geometric hypoelliptic operator, the intrinsic sublaplacian. Due to a degeneracy of its symbol, geometric and analytic effects can be observed in the study of this operator, which have no counterpart in Riemannian geometry. During the last decades inverse spectral problems in subriemannian geometry have been studied by various authors. Typical approaches are based on the analysis of the induced subriemannian heat or wave equation. In this talk we survey some results in subriemannian geometry. In particular, we address the spectral theory of the sublaplacian in the case of certain compact nilmanifolds or, more generally, for $H$-type foliations. This presentation is based on joint work with K. Furutani, C. Iwasaki and A. Laaroussi, I. Markina and G. Vega-Molino.