In this paper, we introduce semi-infinite generalized disjunctive programs that are defined by logical propositions along with disjunctions of sets of logical equations and infinite number of algebraic inequalities. We denote these programs by SIGDPs. For SIGDPs with linear and convex inequalities, we present new reformulations: semi-infinite mixed-binary/disjunctive linear programs and semi-infinite mixed-binary/disjunctive convex programs, respectively. These results are also applicable for solving SIGDPs with nonconvex functions using their convex underestimators. Even for finite GDPs, this leads to introduction of new reformulations that have no big-M parameters and have lesser number of variables, in comparison to reformulations known in the literature for finite GDPs. We also present a tight extended formulation for semi-infinite disjunctive convex programs. Additionally, we study semi-infinite mixed integer convex program after binarizing integer variables (a special case of SIGDP) and present: (a) semi-infinite convex programming equivalent in higher dimensional space, (b) hierarchy of relaxations between the continuous and convex hull of its feasible region, and (c) sequential convexification approach. We showcase the applications of these results for solving robust 0-1 convex programs and distributionally robust convex programs as well. Furthermore, we provide Lagrangian relaxation based approaches embedded with aforementioned results for deriving lower bounds to solve two-stage stochastic and distributionally robust SIGDPs with general ambiguity set.