The authors show that Fefferman's mapping theorem remains true in the category of almost complex manifolds. Theorem 1.1. Let \(D\) and \( D^\prime\) be two smooth relatively compact domains in real four dimensional manifolds. Assume that \(D\) admits an almost complex structure \(J\), smooth on \(\overline{D}\) and such that \((D,J)\) is strictly pseudoconvex. Then a smooth diffeomorphism \(f:D \to D^\prime\) extends to a smooth diffeomorphism between \(\overline{D}\) and \(\overline{D}^\prime\) if and only if the direct image \(f_*(J)\) of \(J\) under \(f\) extends smoothly on \(\overline{D}^\prime \) and \((D^\prime, f_*(J))\) is strictly pseudoconvex. This theorem admits another formulation, closer to the classical one: Theorem 1.2. A biholomorphism between two smooth relatively compact strictly pseudoconvex domains in (real) four dimensional almost complex manifolds extends to a smooth diffeomorphism between their closures. The proof is mainly based on a reflection principle for pseudoholomorphic discs, on precise estimates of the Kobayashi-Royden infinitesimal pseudometric and on the scaling method in almost complex manifolds.