A method of cleaving a double-stranded DNA target recognition sequence in a eukaryotic cell comprising: introducing into said eukaryotic cell a nucleic acid encoding a meganuclease; or a meganuclease protein; wherein said meganuclease cleaves said double-stranded DNA target recognition sequence; and wherein said meganuclease is a recombinant meganuclease comprising: a polypeptide having at least 85% sequence similarity to residues 2-153 of the I-CreI meganuclease of SEQ ID NO: 1; and having specificity for a recognition sequence half-site which differs by at least one base pair from a half-site within an I-CreI meganuclease recognition sequence selected from the group consisting of SEQ ID NO: 2, SEQ ID NO: 3, SEQ ID NO: 4 and SEQ ID NO: 5, wherein specificity at position -1 has been altered: (i) to an A by a modification selected from the group consisting of D75C, D75L, D75Y, K139Y, T46A, and T46C; (ii) to a G by a modification selected from the group consisting of T46E, T46D, and D75E; or (iii) to a T by a modification selected from the group consisting of D75H, D75Q, D75Y, K139H, T46H, and T46Q; (iv) to any base on a sense strand by a T46G modification; with the proviso that the method is not a method for treatment of the human or animal body by therapy; with the proviso that the method is not a method for modifying the germ line genetic identity of human beings and with the further proviso that the cell is not a human embryonic stem cell or human blastocyst cell. introducing into said eukaryotic cell a nucleic acid encoding a meganuclease; or a meganuclease protein; wherein said meganuclease cleaves said double-stranded DNA target recognition sequence; and wherein said meganuclease is a recombinant meganuclease comprising: a polypeptide having at least 85% sequence similarity to residues 2-153 of the I-CreI meganuclease of SEQ ID NO: 1; and having specificity for a recognition sequence half-site which differs by at least one base pair from a half-site within an I-CreI meganuclease recognition sequence selected from the group consisting of SEQ ID NO: 2, SEQ ID NO: 3, SEQ ID NO: 4 and SEQ ID NO: 5, wherein specificity at position -1 has been altered: (i) to an A by a modification selected from the group consisting of D75C, D75L, D75Y, K139Y, T46A, and T46C; (ii) to a G by a modification selected from the group consisting of T46E, T46D, and D75E; or (iii) to a T by a modification selected from the group consisting of D75H, D75Q, D75Y, K139H, T46H, and T46Q; (iv) to any base on a sense strand by a T46G modification; a polypeptide having at least 85% sequence similarity to residues 2-153 of the I-CreI meganuclease of SEQ ID NO: 1; and having specificity for a recognition sequence half-site which differs by at least one base pair from a half-site within an I-CreI meganuclease recognition sequence selected from the group consisting of SEQ ID NO: 2, SEQ ID NO: 3, SEQ ID NO: 4 and SEQ ID NO: 5, wherein specificity at position -1 has been altered: (i) to an A by a modification selected from the group consisting of D75C, D75L, D75Y, K139Y, T46A, and T46C; (ii) to a G by a modification selected from the group consisting of T46E, T46D, and D75E; or (iii) to a T by a modification selected from the group consisting of D75H, D75Q, D75Y, K139H, T46H, and T46Q; (iv) to any base on a sense strand by a T46G modification; (i) to an A by a modification selected from the group consisting of D75C, D75L, D75Y, K139Y, T46A, and T46C; (ii) to a G by a modification selected from the group consisting of T46E, T46D, and D75E; or (iii) to a T by a modification selected from the group consisting of D75H, D75Q, D75Y, K139H, T46H, and T46Q; (iv) to any base on a sense strand by a T46G modification; with the proviso that the method is not a method for treatment of the human or animal body by therapy; with the proviso that the method is not a method for modifying the germ line genetic identity of human beings and with the further proviso that the cell is not a human embryonic stem cell or human blastocyst cell.