We prove the following finite jet determination result for CR mappings: Given a smooth generic submanifold M ⊂ C N , N ⩾ 2 , that is essentially finite and of finite type at each of its points, for every point p ∈ M there exists an integer ℓ p , depending upper-semicontinuously on p, such that for every smooth generic submanifold M ′ ⊂ C N of the same dimension as M, if h 1 , h 2 : ( M , p ) → M ′ are two germs of smooth finite CR mappings with the same ℓ p jet at p, then necessarily j p k h 1 = j p k h 2 for all positive integers k. In the hypersurface case, this result provides several new unique jet determination properties for holomorphic mappings at the boundary in the real-analytic case; in particular, it provides the finite jet determination of arbitrary real-analytic CR mappings between real-analytic hypersurfaces in C N of D'Angelo finite type. It also yields a new boundary version of H. Cartan's uniqueness theorem: if Ω , Ω ′ ⊂ C N are two bounded domains with smooth real-analytic boundary, then there exists an integer k, depending only on the boundary ∂ Ω, such that if H 1 , H 2 : Ω → Ω ′ are two proper holomorphic mappings extending smoothly up to ∂ Ω near some point p ∈ ∂ Ω and agreeing up to order k at p, then necessarily H 1 = H 2 .