There has been much prior work on understanding the complexity of the constraint satisfaction problem (CSP), a broad framework capturing many combinatorial problems. This paper studies a natural and strict generalization of the CSP, the quantified constraint satisfaction problem (QCSP). In the CSP, all variables are existentially quantified; in the QCSP, some variables may be universally quantified. Our contributions include proof systems for the QCSP and the identification of three broad tractable subclasses of the QCSP. Central to our study is the algebraic notion of closure properties of constraints, which has been previously used to study the CSP.