  We define the Hanf number h(L) of a language L to be the least cardinal κ such that, if an L-sentence has a model of cardinality κ, it has models of arbitrarily large cardinality. The existence of h(L) is readily established. For each L-sentence σ not possessing models of arbitrarily large cardinality let κ(σ) be the least cardinal κ such that σ does not have a model of cardinality κ. If λ is the supremum of all the κ(σ), then, if a sentence of L has a model of cardinality λ, it has models of arbitrarily large cardinality.