In the 1960s, R. G. Swan asked for which finite groups G does the integral group ring ZG have the property that P ⊕ ZG ≅ Q ⊕ ZG implies P ≅ Q for projective modules P and Q. Using deep results of Eichler on strong approximation, it was soon shown that this holds provided the Wedderburn decomposition of RG contains no copy of the quaternions H. Much progress was made in the 1970s and 80s to tackle the remaining cases, including Fröhlich’s proof that the cancellation property is closed under quotients of groups and Swan’s classification of when cancellation occurs for binary polyhedral groups. In recent years, several applications to algebraic topology have sparked renewed interest in this problem. In this talk, we will present a general cancellation theorem for projective ZG modules which reduces the problem to determining cancellation for one infinite family as well as 14 exceptional groups.