In this talk I'll review some of the basic concepts of sheaf theory as it is relevant to persistence in one or more parameters. One of the themes that will be emphasized is how the same formalism can be used and simply by changing the underlying topology many different constructs in TDA can be obtained: multi-filtered homology (Alexandrov/gamma topology), level set persistence and Reeb spaces (Euclidean topology), the persistent homology transform and vineyards (mixture of Alexandrov and Euclidean topologies). Additionally, I'll describe how sheaves can be used to organize various counting problems associated to persistence.