Integral equation, simple mode coupling, and activated barrier hopping theories have been combined to study slow dynamics in “biphasic” binary mixtures composed of hard and sticky spheres of identical diameters. Under structural equilibrium conditions the theory has been employed to investigate how the addition of repulsive particles modifies the ideal gelation transition and elastic modulus, and how attractive particles perturb the vitrification of hard spheres. The total volume fraction dependence of the shear modulus is significantly changed relative to its analogous one-component behavior. A two-dimensional nonequilibrium free energy controls particle displacements, and the associated transient localization lengths and barrier hopping processes have been studied as a function of mixture composition and sticky colloid adhesion strength and spatial range. Barriers for activated transport of the attractive and repulsive particles are generally very different, implying a strong dynamical asymmetry. The corresponding quenched porous media problems have also been studied. Ideal colloidal glasses in porous media form at much lower total volume fraction than for the one-component hard sphere fluid. If the sticky particles are mobile in the quenched media, then very rich dynamical behavior is found. For example, mobile colloids may undergo an ideal glass transition, an ideal gel transition, or both. The signature of the latter behavior is a nonequilbrium free energy that acquires two local minima.