  We establish subgeometric bounds on convergence rate of general Markov
processes in the Wasserstein metric. In the discrete time setting we prove that
the Lyapunov drift condition and the existence of a "good" $d$-small set imply
subgeometric convergence to the invariant measure. In the continuous time
setting we obtain the same convergence rate provided that there exists a "good"
$d$-small set and the Douc-Fort-Guillin supermartingale condition holds. As an
application of our results, we prove that the Veretennikov-Khasminskii
condition is sufficient for subexponential convergence of strong solutions of
stochastic delay differential equations.
