Let F be a non-archimedean local field. We establish the local Langlands correspondence for all inner forms of the group SLn(F ). It takes the form of a bijection between, on the one hand, conjugacy classes of Langlands parameters for SLn(F ) enhanced with an irreducible representation of an S-group and, on the other hand, the union of the spaces of irreducible admissible representations of all inner forms of SLn(F ) up to equivalence. An analogous result is shown in the archimedean case.To settle the case where F has positive characteristic, we employ the method of close fields. We prove that this method is compatible with the local Langlands correspondence for inner forms of GLn(F ), when the fields are close enough compared to the depth of the representations. Contents 1. Introduction 1 2. Inner forms of GL n (F ) 5 3. Inner forms of SL n (F ) 9 4. Characterization of the LLC for some representations of GL n (F ) 14 5. The method of close fields 20 6. Close fields and Langlands parameters 26 References 30