In Part I of this monograph we will sketch the 18th Century study, by Euler and others, of what later became known as elliptic integrals. In Part II we will consider the work on inverting certain of these integrals, by Gauss, Abel, and Jacobi, to form elliptic functions. We will consider the work of Weierstrass on his elliptic functions, the work of Riemann on Riemann urfaces, and the work of Klein and others on the elliptic modular function. Until. approximately 1840 the parameters in these integrals were real. In the latter half of the 19th century and during most of the 20th, the real case was largely neglected. The purpose of this monograph is to give a very thorough treatment of the real elliptic case. We will present the theory of real elliptic curves in Part 111. Many of these theorems are new.