As a starting point we study finite-state automata, which represent the simplest devices for recognizing languages. The theory of finite-state automata has been described in numerous textbooks both from a computational and an algebraic point of view. Here we immediately look at the more general concept of a monoidal finite-state automaton, and the focus of this chapter is general constructions and results for finite-state automata over arbitrary monoids and monoidal languages. Refined pictures for the special (and more standard) cases where we only consider free monoids or Cartesian products of monoids will be given later.