An (algebraic) function field is an algebraic extension of the field of univariate rational fractions over a field.
The function field of an algebraic curve defined over a field K is an algebraic function field over K, and every algebraic function field may be obtained in this way.
The importance of this notion relies on the function field analogy which consists in the fact that almost all theorems on number fields have their counterpart on function fields over a finite field, which is frequently easier to prove. In the context of this analogy, number fields and function fields are usually called global fields.
The study of function fields over a finite field has applications in cryptography and error correcting codes. For example, the function field of an elliptic curve (an important mathematical tool for public key cryptography) is an algebraic function field.
Function fields over the field of rational numbers play also an important role in solving inverse Galois problems.