This thesis will study the $f$-vectors and facial structure of two families of flow polytopes: those coming from complete graphs and those coming from grid graphs. In the first part of this thesis, we study flow polytopes coming from the family of complete graphs $K_{n+1}$, which include the famous Chan-Robbins-Yuen polytope ($CRY_n$). We give generating functions and explicit formulas for computing the $f$-vector by using Hille's (2003) result bijecting faces of a flow polytope to certain graphs, as well as Andresen-Kjeldsen's (1976) result that enumerates certain subgraphs of the directed complete graph. We extend our results to flow polytopes of the complete graph having arbitrary (non-negative) netflow vectors and recover the $f$-vector of the Tesler polytope of Mészáros--Morales--Rhoades (2017). In the second part of this thesis we study generalized Pitman-Stanley polytopes ($PS_n^m({\bf a}, {\bf b})$), which we show are flow polytopes of a grid graph for certain netflow vectors. We study the vertices of this family of polytopes, and provide descriptions of such in terms of integer flows. Moreover, we give recursive formulas for the number of faces, as well as generating functions for the number of vertices, and prove that---for a fixed ${\bf a}$ and ${\bf b}$---these generating functions are a polynomial in $m$. In the third part of this thesis, we investigate the face lattice of both of these families of polytopes, giving a recursive description of the face lattice of generalized Pitman-Stanley polytopes when ${\bf b} = {\bf 0}$. Moreover, we investigate a family of polytopes closely related to flow polytopes---fractional perfect matching polytopes---and find a formula for the number of vertices of the fractional perfect matching polytope of the complete graph.