This thesis contains work from the author's papers Palsson and Sovine (2020); Iosevich, Palsson, and Sovine (2022); and Palsson and Sovine (2022) with coauthors Eyvindur Palsson and Alex Iosevich. These works establish new $L^p$-improving, quasi-Banach, and sparse bounds for several bilinear and multilinear operators that generalize the linear spherical average to the multilinear setting, and maximal variants of these operators, with an emphasis on the triangle averaging operator and the bilinear spherical averaging operator. / Doctor of Philosophy / This thesis establishes new regularity properties for several mathematical operations that generalize the operation of taking the average of a function over a sphere to operations that average the product of several input functions over a surface to produce a single output function. These operations include the triangle averaging operator, the $k$-simplex averaging operators for $k$ an integer greater than 1, and the bilinear spherical averaging operator, as well as maximal operators obtained by allowing the radius of the averaging surface to vary over some range of values.