We give a self-contained proof of a fundamental result of Makanin (1977), which solves the satisfiability problem of equations with constants over free monoids. Our presentation of Makanin's algorithm is borrows Schulz (1992a), where Makanin's result is extended to the case where solutions are restricted by imposing regular constraints on the variables. This report appears (with minor modifications) as a chapter of the new book of M. Lothaire Algebraic Combinatorics on Words.
