One of the less known characterizations of a Ramsey cardinal kappa involves the existence of certain types of elementary embeddings for transitive sets of size kappa satisfying a large fragment of ZFC. I introduce new large cardinal axioms by isolating and generalizing the key properties of Ramsey embeddings and show that they form a natural hierarchy between weakly compact cardinals and measurable cardinals. The stronger of these large cardinal notions are better suited than Ramsey cardinals for indestructiblity arguments. The weaker of the new large cardinals further our knowledge about the elementary embedding properties of smaller large cardinals, in particular those still consistent with V=L. A large portion of this work is joint with Philip Welch.