A rapid method for generating a set of discrete DNA amplification products characteristic of a genome as a "fingerprint" comprises the steps of: priming target nucleic acid of a genome with an single-stranded primer to form primed nucleic acid such that a substantial degree of internal-mismatching occurs between the primer and the target nucleic acid; amplifying the primed nucleic acid by performing at least one cycle of polymerase chain reaction amplification; and amplifying the product of step (2) by performing at least about 10 cycles of polymerase chain reaction amplification. The method is known as the arbitrarily primed polymerase chain reaction (AP-PCR) method and is suitable for the identification of bacterial species and strains, including Staphylococcus and Streptococcus species, mammals and plants. The method of the present invention can identify species rapidly, using only a small amount of biological material, and does not require knowledge of the nucleotide sequence or other molecular biology of the nucleic acids of the organisms to be identified. Only one primer sequence is required for amplification and/or identification. The method can also be used to generate detectable polymorphisms for use in genetic mapping of animals and humans.