Solvability is a key notion in the theory of call-by-name λ-calculus, used in particular to identify meaningful terms. However, adapting this notion to other call-by-name calculi, or extending it to different models of computation —such as call-by-value—, is not straightforward. In this paper, we study solvability for call-by-name and call-by-value λ-calculi with generalized applications, both variants inspired from von Plato’s natural deduction with generalized elimination rules. We develop an operational as well as a logical theory of solvability for each of them. The operational characterization relies on a notion of solvable reduction for generalized applications, and the logical characterization is given in terms of typability in an appropriate non-idempotent intersection type system. Finally, we show that solvability in generalized applications and solvability in the λ-calculus are equivalent notions.