A method for enriching for target sequences which contain sequence variants over wild-type target sequences comprising: (a) exposing the target sequences and reference sequences to a reaction mixture containing pairs of nucleic acid primers, each pair of primers generating under nucleic acid amplification conditions an amplicon that is a copy of a target sequence and a reference sequence, wherein the amplicon generated by one pair of primers is different from the amplicon generated from each other pair of primers, (b) subjecting the target sequences and the reference sequences in the reaction mixture to a denaturing temperature that is above the melting temperature of the target sequences and the reference sequences, (c) reducing the temperature of the reaction mixture to permit the formation of a target strand/reference strand duplexes, (d) subjecting the duplexes in the reaction mixture to a first critical denaturation temperature Tc that is below the lowest Tm of any duplex formed by a wild-type target sequence and a reference sequence, to permit selective denaturation of the duplexes containing sequence variants, without denaturation of the duplexes formed by wild-type target sequences with the reference sequences, (e) reducing the temperature of the reaction mixture in the presence of pairs of nucleic acid primers and permitting the primers to anneal to the target sequences, (f) extending the primers to enrich the target sequences, (g) repeating steps (d) to (f) at least once at a second critical denaturation temperature which is above the first critical denaturation temperature and below the melting temperature Tm of the duplex formed by second wild-type target sequence and second reference sequence to ensure selective amplification of all target sequences that contain sequence variants over wild type sequences. (a) exposing the target sequences and reference sequences to a reaction mixture containing pairs of nucleic acid primers, each pair of primers generating under nucleic acid amplification conditions an amplicon that is a copy of a target sequence and a reference sequence, wherein the amplicon generated by one pair of primers is different from the amplicon generated from each other pair of primers, (b) subjecting the target sequences and the reference sequences in the reaction mixture to a denaturing temperature that is above the melting temperature of the target sequences and the reference sequences, (c) reducing the temperature of the reaction mixture to permit the formation of a target strand/reference strand duplexes, (d) subjecting the duplexes in the reaction mixture to a first critical denaturation temperature Tc that is below the lowest Tm of any duplex formed by a wild-type target sequence and a reference sequence, to permit selective denaturation of the duplexes containing sequence variants, without denaturation of the duplexes formed by wild-type target sequences with the reference sequences, (e) reducing the temperature of the reaction mixture in the presence of pairs of nucleic acid primers and permitting the primers to anneal to the target sequences, (f) extending the primers to enrich the target sequences, (g) repeating steps (d) to (f) at least once at a second critical denaturation temperature which is above the first critical denaturation temperature and below the melting temperature Tm of the duplex formed by second wild-type target sequence and second reference sequence to ensure selective amplification of all target sequences that contain sequence variants over wild type sequences.