An argument is a reason or justification of a claim. It has an intrinsic strength and may be attacked by other arguments. Hence, the evaluation of its overall strength becomes mandatory, especially for judging the reliability of its claim. Such an evaluation is done by acceptability semantics. The aim of this paper is to set up the foundations of acceptability semantics. Foundations are important not only for a better understanding of the evaluation process in general, but also for clarifying the basic assumptions underlying semantics, for comparing different (families of) semantics and identifying families of semantics that have not been explored yet. The paper defines the building blocks of a semantics. It introduces key concepts and principles on which an evaluation is based. Each concept (principle) is described by an axiom. We investigate properties of semantics that satisfy the axioms, show the foundations of the two crucial notions of reinstatement and defence, and analyse some existing semantics against the axioms.