My project is in the area of Commutative Algebra and its interactions with Algebraic Geometry, Combinatorics, and Convex Geometry. More precisely, the main goal is to study several (algebraic, geometrical and combinatorial) features of the notion of multidegrees or mixed multiplicities. The concept of multidegree provides the right generalization of degree to a multiprojective setting, and its study goes back to seminal work by van der Waerden in 1929. Since then, multidegrees have appeared in many results and applications that interconnect various branches of mathematics. This project proposal is aimed to prove highly original results that will improve our understanding of multidegrees (or mixed multiplicities). My first goal is to define and develop the notion of multidegree or mixed multiplicity in more general settings. Then, a second goal is to find explicit characterizations for important properties of multidedgrees (e.g., determining when multidegrees are positive). As a particular case, I plan to study the projective degrees of certain rational maps. My third goal is to define and develop the notion of mixed multiplicities for (not necessarily Noetherian) graded families of ideals.