  Under natural assumptions, we prove the ergodicities and exponential
ergodicities in Wasserstein and total variation distances of Dawson--Watanabe
superprocesses without or with immigration. The strong Feller property in the
total variation distance is derived as a by-product. The key of the approach is
a set of estimates for the variations of the transition probabilities. The
estimates in Wasserstein distance are derived from an upper bound of the
kernels induced by the first moment of the superprocess. Those in total
variation distance are based on a comparison of the cumulant semigroup of the
superprocess with that of a continuous-state branching process. The results
improve and extend considerably those of Stannat (2003a, 2003b) and Friesen
(2019+). We also show a connection between the ergodicities of the associated
immigration superprocesses and decomposable distributions.
