  Elementary embeddings are natural maps to consider within first-order model theory. Around 1950 Abraham Robinson was impressed that maps between algebraic structures in general seem hardly ever to be elementary, whereas some important maps (such as embeddings between two algebraically closed fields, or between two real-closed fields) turn out to be elementary. He was also surprised to find that this fact about algebraically closed fields is another way of stating a celebrated theorem called the Hilbert Nullstellensatz. These observations of Robinson have had a huge effect on the development of model theory. In Robinson’s terminology, a first-order theory is model-complete if every embedding between models of the theory is elementary. This notion has found many uses, and it often appears in applications of model theory in algebra.