Molecular integral equation, simplified mode coupling, and activated barrier hopping theories have been combined to systematically investigate the structure, slow translational dynamics, vitrification and gelation of suspensions composed of nonspherical colloids. One (rod), two (planar) and three (e.g. cube, octahedron) dimensional shapes have been studied. For purely excluded volume interactions, the effective dimensionality of the rigid object is the primary variable controlling the ideal glass volume fraction and entropic barrier height. Subtle effects occur within a fixed dimensional class (e.g., triangles versus hexagons). In the presence of strong, short range inter-site attractions, glass formation competes with gelation, and re-entrant glass-fluid-gel behavior is predicted at high volume fractions, the quantitative aspects of which depend on colloidal shape. Correlations of the slow dynamics with the packing, and number of sticky contacts between the nonspherical objects, has been investigated. All the results for molecular colloids are contrasted with their spherical particle analogs to establish the consequences of shape anisotropy. Finally, a new unified understanding of how barriers in colloidal gels composed of sticky spheres depend on local structure, attraction strength, and volume fraction has been achieved. The emergent physical picture suggests analogies with glasses and mechanically jammed systems.