  This article is an exposition of an elementary constructive proof of
canonical resolution of singularities in characteristic zero, presented in
detail in Invent. Math. 128 (1997), 207-302. We define a new local invariant
and get an algorithm for canonical desingularization by successively blowing up
its maximum loci. The invariant can be described by local computations that
provide equations for the centres of blowing up. We describe the origin of our
approach and present the proof (in the hypersurface case) in parallel with a
worked example.
