This paper presents semantic specifications and correctness proofs for both on-line and offline partial evaluation of strict first-order functional programs. To do so, our strategy consists of defining a core semantics as a basis for the specification of three non-standard evaluations: instrumented evaluation, on-line and off-line partial evaluation. We then use the technique of logical relations to prove the correctness of both on-line and off-line partial evaluation semantics. The contributions of this work are as follows. 1. We provide a uniform framework to defining and proving correct both on-line and off-line partial evaluation. 2. This work required a formal specification of on-line partial evaluation with polyvariant specialization. We define criteria for its correctness with respect to an instrumented standard semantics. As a byproduct, on-line partial evaluation appears to be based on a fixpoint iteration process, just like binding-time analysis. 3. We show that binding-time...