I explain some cancellation and non-cancellation phenomena in algebraic geometry, such as the Gromov cancellation question: If the open complements of two closed subvarieties of an ambient variety are isomorphic, are these closed subvarieties birational? This question has a positive answer when varieties are smooth or simple normal crossing, and a negative answer in more general cases. The emphasis will be on the structure of the Grothendieck ring of varieties and on the groups of birational self-maps of algebraic varieties. This will be an overview of the work of many people in the last 25 years since Gromov's original formulation, including Larsen-Lunts, Borisov, Zakharevich, Hassett-Lai, Kontsevich-Tschinkel, and most recently, work by Lin and myself.