A quandle is a set with a right invertible, self distributive, binary
operation for which every element is an idempotent. For example, consider
a group under conjugation. Classical and higher dimensional knots have a
fundamental quandle. The fundamental quandle is a stronger notion than
that of the fundamental group.
There is a homology theory of quandles that parallels the homology theory of groups. From this homology theory we can derive invariants of classical knots and knotted surfaces that are not found in the fundamental quandle itself.
In this talk, I will sketch the homology and cohomology theory of quandles, and illustrate how the theory is being applied by various authors.