  We provide simple and constructive proofs of Harris-type theorems on the
existence and uniqueness of an equilibrium and the speed of equilibration of
discrete-time and continuous-time stochastic semigroups. Our results apply both
to cases where the relaxation speed is exponential (also called geometric) and
to those with no spectral gap, with non-exponential speeds (also called
subgeometric). We give constructive estimates in the subgeometric case and
discrete-time statements which seem both to be new. The method of proof also
differs from previous works, based on semigroup and interpolation arguments,
valid for both geometric and subgeometric cases with essentially the same
ideas. In particular, we present very simple new proofs of the geometric case.
