  We present a general coalgebraic setting in which we define finite and
infinite behaviour with B\"uchi acceptance condition for systems whose type is
a monad. The first part of the paper is devoted to presenting a construction of
a monad suitable for modelling (in)finite behaviour. The second part of the
paper focuses on presenting the concepts of a (coalgebraic) automaton and its
($\omega$-) behaviour. We end the paper with coalgebraic Kleene-type theorems
for ($\omega$-) regular input. The framework is instantiated on
non-deterministic (B\"uchi) automata, tree automata and probabilistic automata.
