The paper is devoted to the study of a new class of optimization problems with objectives given as differences of convex (DC) functions and constraints described by infinitely many convex inequalities. We consider such problems in the general framework of locally convex topological vector spaces, although the major results obtained in the paper are new even in finite dimensions when the problems under consideration reduce to DC semi-infinite programs. The main attention is paid to deriving qualified necessary optimality conditions as well as necessary and sufficient optimality conditions for DC infinite and semi-infinite programs and to establishing relations between various qualification conditions. The results obtained are applied to and specified for particular classes of DC programs involving polyhedral convex functions in DC objectives, programs with cone constraints as well as those with positive semi-definite constraints described via the Löwner partial order.