Subriemannian (SR) geometry provides a mathematical model for motion under non-holonomic constraints. Concrete problems in mathematics, physics or applications can often be formulated in the Subriemannian context. After providing the basic definitions we recall some examples and constructions of SR geometries. Relevant questions concern horizontal connectivity as well as the classification of SR geodesics and exponential map. From an analytic point of view we recall the construction of an induced geometric differential operators called the (intrinsic) sub-Laplacian which fulfills subelliptic estimates and therefore is hypoelliptic. The corresponding SR heat equation and its fundamental solution K (SR heat kernel) establish a link between geometry and analysis and in some cases K encodes the spectrum of the operator. If available, one is interested in explicit formulas for the heat kernel. In the case of a compact manifold, eigenvalue asymptotics or the inverse spectral problem in this non-elliptic configuration can be studied.