Dold sequences constitute an important class of integer sequences that play an important role in combinatorics, number theory, topology and dynamical systems. We generalize the notion of Dold sequence for the case of partially ordered sets and describe their properties. In particular we give two alternative descriptions of generalized Dold sequences: by some class of elementary sequences as well as by different generalized arithmetical functions, both defined on a partially ordered set. We also define vector Dold sequences and show their combinatorial interpretation in terms of periodic points.