  This monograph, written for educational purposes, serves as an introduction
to the concept of integrability as it applies to systems of differential
equations (both ordinary and partial) as well as to vector-valued fields. The
general cases of integrability examined are: path-independence of line
integrals of vector fields on the plane and in space; integration of systems of
ordinary differential equations by using first integrals; and integrable
systems of partial differential equations. Special topics include the
integration of analytic functions on the complex plane and some elements from
the geometric theory of differential systems. Certain more advanced subjects,
such as Lax pairs and B\"acklund transformations, are also discussed.
