In this paper we study coalgebras, that is, vector spaces A with a linearmap A→A⊗A. To every coalgebra A one can associate an algebra on its dual A*such that A becomes a bimodule for A*. We exploit this construction to define the concept of a variety of coalgebras unifying under a general framework the known examples of associative and Lie coalgebras: A is a coalgebra of the variety θ if A*is an algebra in θ. On the other hand we study structural properties of coalgebras by translating them into the language of algebras. In particular we prove the analogue of the Fundamental Theorem on associative coalgebras Q]: alternative and Jordan coalgebras are locally finite