Equicontinuity-type concepts for random functions, which are important for establishing convergence results for such functions, have increasingly been used in the econometrics literature. In this paper we define and discuss various equicontinuity-type concepts for random functions and employ those concepts to provide sufficient conditions for uniform convergence and, in particular, for uniform laws of large numbers. Furthermore, we clarify the differences and similarities between uniform laws of large numbers based on pointwise and local laws of large numbers given in the recent literature as they relate to differences m the employed equicontinuity-type concepts.