The resolution of singularities has a long history dating back to Newton and his work with plane curves. The problem for higher-dimensional singular varieties in characteristic 0 wasn't solved until Hironaka's famous paper of 1964. His proof is said to be one of the longest and most difficult in mathematics, and highly non-constructive. More constructive and algorithmic proofs were later found by mathematicians such as Bierstone and Milman.

Given a singular algebraic variety, we would like to find a smooth variety which is birationally equivalent to it. To do this, we will need to use a birational map called a blowing up. To know what to do at each step of the algorithm, a desingularization invariant will be introduced which both computes the next centre of blowing up, and measures improvements in the singularities.

I will give an introduction to the problems associated with resolving singularities, and I will try to explain some of the key aspects of the resolution algorithm.