Finite Element Computational Gas Dynamics: Multi-Dimensional Upstream Biased Formulation details a systematic characteristics-based finite element procedure to investigate compressible inviscid and viscous inert and chemically reacting laminar and turbulent flows. The book is structured in two major parts: I) the continuum world and II) the discretized environment. Part I details the development of the governing systems from first thermomechanical principles and establishes an inherently multidimensional upstream bias formulation directly at the differential equation level, before any discretization. Part II presents grid generation techniques that use algebraic and differential methods. These are followed by an optimal finite element formulation that incorporates efficient metric data modelling, high-order Chebyshev elements, and accurate enforcement of boundary conditions. It is also shown that the formulation expands upon the Discontinuous Galerkin Method. The scalability of the overall algorithm is covered in dedicated sections of the book. The overall finite element formulation is then shown to generate accurate essentially non-oscillatory solutions that crisply capture contact discontinuities and normal as well as oblique shocks, for transonic, supersonic, and hypersonic inviscid and viscous as well as steady and unsteady laminar and turbulent flows. This book is for graduate and upper-level undergraduate students as well as scientists and engineers in Aerospace Engineering, Mechanical Engineering, Physics, and Mathematics.