Algebraic geometry is the elaboration of Descartes' insight that problems in geometry may be converted to questions about systems of algebraic equations and, conversely, by thinking in terms of sets of solutions, problems about polynomial equations may be regarded geometrically.

Traditionally, algebraic geometry concerns the properties affine complex algebraic varieties: sets of the form

Complex algebraic varieties may be regarded as topological, and even analytic spaces, with the structure inherited from the ambient Euclidean space. As such, smooth complex algebraic varieties are complex manifolds and, in general, for any complex algebraic variety X there is a finite sequence

Algebraic geometry progressed from the study of complex algebraic varieties to the study of algebraic varieties over more general algebraically closed fields, in which the field of complex numbers is replaced by a general algebraically closed field, or just a field, or even a general commutative ring. With the theory of schemes, the relation between geometry in its naïve sense and algebraic geometry becomes less immediate but deeper, as, for instance, many constructions of algebraic topology and differential geometry may be performed cleanly and strictly algebraically.

To the extent that algebraic geometry concerns algebraic varieties over algebraically closed fields, or even algebraic varieties over fields, ACF serves as an excellent framework for the model-theoretic study of algebraic geometry.