  In this paper, we develop the theory of convolutional codes over finite
commutative chain rings. In particular, we focus on maximum distance profile
(MDP) convolutional codes and we provide a characterization of these codes,
generalizing the one known for fields. Moreover, we relate MDP convolutional
codes over a finite chain ring with MDP convolutional codes over its residue
field. Finally, we provide a construction of MDP convolutional codes over
finite chain rings generalizing the notion of superregular matrices.
