Production correspondences are defined and their properties explored. Properties of the distance function of a production structure are studied and are related to properties of the correspondence. Homotheticity is generalized to allow discontinuities in returns to scale and to take cognizance of several outputs. The factorization of the distance function into the product of two functions, one depending only on output and the other only on input, is demonstrated. The cost function is then shown to give rise to a production structure. A Shephard-type duality theorem is proven which implies, among other things, that the cost function and the distance function are dually related; that is, given one, a minimization proglem yields the other. The duality theorem is then used to deduce necessary and sufficient conditions, in terms of the cost function, for a production correspondence to be homothetic. A few simple applications of the duality theorem are given which relate properties of the cost function to geometric properties of the correspondence.