A concept of equation morphism is introduced for every endofuctor $F$ of a cocomplete category $\Ce$. Equationally defined classes of $F$-algebras for which free algebras exist are called varieties. Every variety is proved to be monadic over $\Ce$, and, conversely, every monadic category is equivalent to a variety. The Birkhoff Variety Theorem is also proved for $`{\sf Set}\hbox{-like}'$ categories.

By dualising, we arrive at a concept of coequation such that covarieties, that is, coequationally specified classes of coalgebras with cofree objects, correspond precisely to comonadic categories. Natural examples of covarieties are presented.