The functions studied in this monogra9h are a cross between elliptic functions and modular forms in one variable. Specifically, we define a Jacobi form on SL ( ) to be a holomorphic function 2 (JC = upper half-plane) satisfying the t\-10 transformation eouations 2Tiimcz- k CT ]d a-r +b z ) (1) ( (cT+d) e cp(T, z) cp CT +d ' CT +d (2) rjl(T, z+h+]l) and having a Four-ier expansion of the form 00 e2Tii(nT +rz) (3) cp(T, z) 2: c(n, r) 2:: rE n=O 2 r 4nm Here k and m are natural numbers, called the weight and index of rp, respectively. Note that th e function cp (T, 0) is an ordinary modular formofweight k, whileforfixed T thefunction z-+rjl(-r, z) isa function of the type normally used to embed the elliptic curve / T + into a projective space. If m= 0, then cp is independent of z and the definition reduces to the usual notion of modular forms in one variable. We give three other examples of situations where functions satisfying (1)-(3) arise classically: 1. Theta series. Let Q: -+ be a positive definite integer valued quadratic form and B the associated bilinear form.