When is length a length function?

We show that ideal-length defines a length function on almost-Noetherian integral domains. This length function is a sum of multiplicities and in favourable cases, a linear combination on N of discrete valuations. As a consequence, Σ1-Noetherian domains have length functions. Infra-Krull domains are weakly Krull almost-Noetherian domains. We characterize these integral domains and establish their properties, placing emphasis on integral closures and length functions. Descent properties are shown. Applications to the computation of elasticities and factorization properties are given.