In this paper we discuss duality theory of optimization problems with a linear objective function and subject to linear constraints with cone inclusions, referred to as conic linear problems. We formulate the Lagrangian dual of a conic linear problem and survey some results based on the conjugate duality approach where the questions of ``no duality gap" and existence of optimal solutions are related to properties of the corresponding optimal value function. We discuss in detail applications of the abstract duality theory to the problem of moments, linear semi-infinite and continuous linear programming problems.