Let $M$ be a quasiprojective algebraic manifold with $K_M=0$ and $G$ a finite automorphism group of $M$ acting trivially on the canonical $K_M$ ; for example, a subgroup $G$ of $\operatorname {SL}(n,\mathbb C)$ acting on $\mathbb C^n$ in the obvious way. We aim to study the quotient variety $X=M/G$ and its resolutions $Y\to X$ (especially under the assumption that $Y$ has $K_Y=0$) in terms of $G$-equivariant geometry of $M$. At present we know $4$ or $5$ quite different methods of doing this, taken from string theory, algebraic geometry, motives, moduli, derived categories, etc. For $G$ in $\operatorname {SL}(n,\mathbb C)$ with $n=2$ or $3$, we obtainseveral methods of cobbling together a basis of the homology of $Y$ consisting of algebraic cycles in one-to-one correspondence with the conjugacy es or the irreducible representations of $G$.