Disclosed herein is a composition in the form of an emulsion for shaping or altering the shape of hair, such as by straightening hair, wherein the composition contains a reducing agent, a neutralizing agent, at least two fatty substances, a nonionic surfactant selected from alkoxylated fatty alcohols and alkyl(ether)phosphates, optionally a quaternary ammonium compound, and water, wherein the pH of the composition ranges from about 2 to less than about 7. Also disclosed is a process for shaping or altering the shape of hair.