In this chapter we develop some algebraic tools needed for the general theory of representations and invariants. The central result is a duality theorem for locally regular representations of a reductive algebraic group G. The duality between the irreducible regular representations of G and irreducible representations of the commuting algebra of G plays a fundamental role in classical invariant theory. We study the representations of a finite group through its group algebra and characters, and we construct induced representations and calculate their characters.