An essential notion in semantics is that of logical consequence. A sentence P is a logical consequence of a set of sentences S if and only if it is not possible for the sentences in S to be true and the sentence P to be false at the same time. This is basically the definition (although given in the more abstract terminology of models) that Tarski gave of logical consequence in 1936. In the last twenty years there has been a heated debate on the exact nature of Tarski's theory of logical consequence. This debate has important consequences for the philosophical problem of giving an exact characterization of this central notion in philosophy. Moreover, the intense focus on the original publication by Tarski has led to several disagreements with respect to important interpretative issues related to Tarski's contribution. This chapter provides evidence to show that in 1936 (and 1940) Tarski defended a notion of logical consequence (based on a fixed domain conception) that is at odds with the model-theoretic one common today. The addendum to the chapter defends the interpretation against recent criticism by Gomez-Torrente.