This talk will be divided into three parts.

Part one looks at definitions and properties of critical sets in Latin squares, results on the spectrum of critical sets and the structure of critical sets.

Part two deals with known results on cycle systems, recent results on Alspach's conjecture and the Oberwolfach problem.

Part three discusses the algebraic relations between Latin squares and cycle systems. The talk will conclude by giving some results on single laws for the varieties of groupoids (quasigroups) associated with cycle systems.

Convenor: Simon Clarke.