In model theory, a first-order theory is called **model complete** if every embedding of its models is an elementary embedding.
Equivalently, every first-order formula is equivalent to a universal formula.
This notion was introduced by Abraham Robinson.

A **companion** of a theory *T* is a theory *T** such that every model of *T* can be embedded in a model of *T** and vice versa.

A **model companion** of a theory *T* is a companion of *T* that is model complete. Robinson proved that a theory has at most one model companion. Not every theory is model-companionable, e.g. theory of groups. However if *T* is an -categorical theory, then it always has a model companion.^{[1]}^{[2]}

A **model completion** for a theory *T* is a model companion *T** such that for any model *M* of *T*, the theory of *T** together with the diagram of *M* is complete. Roughly speaking, this means every model of *T* is embeddable in a model of *T** in a unique way.

If *T** is a model companion of *T* then the following conditions are equivalent:^{[3]}

*T** is a model completion of*T**T*has the amalgamation property.

If *T* also has universal axiomatization, both of the above are also equivalent to:

*T** has elimination of quantifiers

- Any theory with elimination of quantifiers is model complete.
- The theory of algebraically closed fields is the model completion of the theory of fields. It is model complete but not complete.
- The model completion of the theory of equivalence relations is the theory of equivalence relations with infinitely many equivalence classes, each containing an infinite number of elements.
- The theory of real closed fields, in the language of ordered rings, is a model completion of the theory of ordered fields (or even ordered domains).
- The theory of real closed fields, in the language of rings, is the model companion for the theory of formally real fields, but is not a model completion.

- The theory of dense linear orders with a first and last element is complete but not model complete.
- The theory of groups (in a language with symbols for the identity, product, and inverses) has the amalgamation property but does not have a model companion.

If *T* is a model complete theory and there is a model of *T* that embeds into any model of *T*, then *T* is complete.^{[4]}