  We prove exponential convergence to the invariant measure, in the total
variation norm, for solutions of SDEs driven by $\alpha$-stable noises in
finite and in infinite dimensions. Two approaches are used. The first one is
based on Harris theorem, and the second on Doeblin's coupling argument.
Irreducibility, Lyapunov function techniques, and uniform strong Feller
property play an essential role in both approaches. We concentrate on two
classes of Markov processes: solutions of finite-dimensional equations,
introduced in [Priola 2010], with H\"older continuous drift and a general,
non-degenerate, symmetric $\alpha$-stable noise, and infinite-dimensional
parabolic systems, introduced in [Priola-Zabczyk 2009], with Lipschitz drift
and cylindrical $\alpha$-stable noise. We show that if the nonlinearity is
bounded, then the processes are exponential mixing. %under the total variation
norm. This improves, in particular, an earlier result established in
[Priola-Xu-Zabczyk 2010] using the weak convergence induced by the
Kantorovich-Wasserstein metric.
