New families of generalized functions are introduced (with an "integral type" evaluation of the behavior of representatives). A strong "average" (on the parameter) association with distributions is introduced (along an analogous line to Cesaro convergence of sequences), and comparisons are made with usual regularities of distributions (only the case of ${\cal{C}} ^{\infty}$ and ${\cal{G}} ^{\infty}$ type regularities is treated in detail here).