given any complex power series in n-variables one can always ask what is the largest negative power which is integrable. this number is the log canonical threshold (its reciprocal is called the arnold multiplicity) of the underlying hypersurface. it is a measure of the complexity of the singularity at the origin, which carries more information than the multiplicity. i describe some recent work with hacon and xu, where we prove some conjectures of kollár and shokurov, which state that the set of log canonical thresholds satisfies the ascending chain condition and which identifies the accumulation points.