In phylogenetics, a taxon is polyphyletic (Greek for "of many races") if the trait its members have in common evolved separately in different places in the phylogenetic tree. Equivalently, a polyphyletic taxon does not contain the most recent common ancestor of all its members.
For example, the group of warm-blooded animals is polyphyletic, because it contains both mammals and birds, but the most recent common ancestor of mammals and birds was cold-blooded. Warm-bloodedness evolved separately in the ancestors of mammals and the ancestors of birds, so it is not a true phylogenetic grouping.
Scientific classification aims to group species together such that every group is descended from a single common ancestor, and the elimination of groups that are found to be polyphyletic is therefore a common goal, and is often the stimulus for major revisions of the classification schemes. A polyphyletic group can be "fixed" either by excluding clades or by adding the common ancestor.
Opinions differ as to whether valid groups need to contain all the descendants of a common ancestor. Groups that do so are called monophyletic, and according to cladistics it should be the aim of classification to ensure that all groups have this property. However, many other taxonomists would argue that there is a valid place for groups that are paraphyletic, i.e. contain only the descendants of a common ancestor, but do not contain all its descendants.