Bordism groups are groups of bordism classes of compact manifolds, with disjoint union as the group operation. They include ordinary bordism groups, also called cobordism groups or cobordism rings, and singular bordism groups. The ordinary bordism groups give a framework for the question, "When is a compact boundaryless manifold the boundary of another manifold?" The answer is, precisely when all its Stiefel-Whitney numbers are zero. Singular bordism groups give insight into Steenrod's realization problem: "When can homology classes be realized as the image of fundamental classes of manifolds?" That answer is known, too.
The machinery of the bordism group winds up being important for homotopy theory as well.
