  The Lov\'asz Local Lemma is a powerful probabilistic technique for proving
the existence of combinatorial objects. It is especially useful for colouring
graphs and hypergraphs with bounded maximum degree. This paper presents a
general theorem for colouring hypergraphs that in many instances matches or
slightly improves upon the bounds obtained using the Lov\'asz Local Lemma.
Moreover, the theorem directly shows that there are exponentially many
colourings. The elementary and self-contained proof is inspired by a recent
result for nonrepetitive colourings by Rosenfeld [2020]. We apply our general
theorem in the setting of proper hypergraph colouring, proper graph colouring,
independent transversals, star colouring, nonrepetitive colouring, frugal
colouring, Ramsey number lower bounds, and for $k$-SAT.
