We study the counterparts of conic linear programs, i.e., problems of optimization of linear functions on intersections of a convex cone with an affine subspace, in a projective setting. The projective theory is in many respects more neat and symmetric than the affine one. The classification of projective programs with respect to boundedness and feasibility is simpler and more transparent, providing interpretations which are obscured in the affine setting. A central feature of this classification is the equivalence between infeasibility of the primal program and the appearance of singularities in the dual program and vice versa. The cost function and the linear constraint cannot anymore be separated and fuse into a single object. Since infinity is an ordinary point on the projective line, infinite values of the objective function are no more conceptually different from finite ones. This erases in some cases the difference between bounded and unbounded problem instances and reveals symmetries which were hidden in the affine setting.