Some of the fundamental notions of linear algebra are the concepts of a generator and a basis for a vector space. In the category theoretic formulation of universal algebra, vector spaces are the Eilenberg-Moore algebras over the free vector space monad on the category of sets. In this paper we investigate general notions of generators and bases for algebras over arbitrary monads on arbitrary categories. On the one hand, we establish purely algebraic results, for instance about the existence and uniqueness of generators and bases, and the representation of algebra homomorphisms. On the other hand, we use the general notion in the context of coalgebraic systems and show that a generator for the underlying algebra of a bialgebra gives rise to an equivalent free bialgebra. As a result, we are able to recover known constructions from automata theory such as the canonical residual finite state automaton and the minimal xor automaton. Finally, we instantiate the framework to a variety of example monads, both set and non set-based.