The antigen receptors on cells of innate immune systems recognize broadly expressed markers on non-host cells while the receptors on lymphocytes of the adaptive immune system display a higher level of specificity. Adaptive immunity, with its exquisite specificity and immunological memory, has only been found in the jawed vertebrates, which also display innate immunity. Jawless fishes and invertebrates only have innate immunity. In the adaptive immune response, T and B-lymphocytes detect foreign agents or antigens using T cell receptors (TCR) or immunoglobulins (Ig), respectively. While Ig can bind free intact antigens, TCR only binds processed antigenic fragments that are presented on molecules encoded in the major histocompatibility complex (MHC). MHC molecules display variation through allelic polymorphism. A diverse repertoire of Ig and TCR molecules is generated by gene rearrangement and junctional diversity, processes carried out by the recombinase activating gene (RAG) products and terminal deoxynucleotidyl transferase (TdT). Thus, the molecules that define adaptive immunity are TCR, Ig, MHC molecules, RAG products and TdT. No direct predecessors of these molecules have been found in the jawless fishes or invertebrates. In contrast, the complement cascade can be activated by either adaptive or innate immune systems and contains examples of molecules that gradually evolved from non-immune functions to being part of the innate and then adaptive immune system. In this paper we examine the molecules of the adaptive immune system and speculate on the existence of direct predecessors that were part of innate immunity.