The paper discusses several first-order modal logics that extend the classical predicate calculus. The model theory involves possible worlds with world-variable domains. The logics rely on the philosophical tenet known as serious actualism in that within modal contexts they allow existential generalization from atomic formulas. The language may or may not have a sign of identity, includes no primitive existence predicate, and has individual constants. Some logics correspond to various standard constraints on the accessibility relation, whereas others correspond to various constraints on the domains of the worlds. Soundness and strong completeness are proved in every case; a novel method is used for proving completeness.