The paper presents a new theoretical approach to the design and understanding of finite element discretizations for a wide variety of systems of partial differential equations (PDEs), known as finite element exterior calculus. This approach utilizes tools from differential geometry, algebraic topology, and homological algebra to develop discretizations that are compatible with the geometric, topological, and algebraic structures underlying well-posedness of PDE problems. The key idea is to view finite element spaces as spaces of piecewise polynomial differential forms, which connect to each other in discrete subcomplexes of elliptic differential complexes. These spaces are also related to the continuous elliptic complex through projections that commute with the complex differential. Applications of this approach include the finite element discretization of the Hodge Laplacian, Maxwell's equations, the equations of elasticity, and elliptic eigenvalue problems, as well as the construction of preconditioners. The paper also discusses the importance of stability in finite element methods, which is closely related to the well-posedness of the discrete equations. Stability is a subtle matter and can be very challenging to achieve, especially for important problems. The paper introduces the concept of finite element exterior calculus, which involves the development of finite element subcomplexes of certain elliptic differential complexes and cochain projections onto them. These subcomplexes inherit the cohomology and other features of the exact complexes. The main theme of the paper is the development of finite element subcomplexes of certain elliptic differential complexes and cochain projections onto them, and their implications and applications in numerical PDEs. The paper also discusses the computational challenges that motivated the development of finite element exterior calculus, including the poor behavior of seemingly reasonable numerical methods and the development of effective methods. The finite element exterior calculus provides an explanation for the difficulties experienced with naive methods and also points to a practical finite element solution. The paper presents a detailed discussion of the finite element exterior calculus, including the development of finite element spaces, the use of the Koszul complex, and the application of the Hodge star operation. It also discusses the use of group representation theory to characterize the spaces of polynomial differential forms used to construct finite element differential forms. The paper concludes with a discussion of the applications of finite element exterior calculus to various problems, including the Hodge Laplacian, eigenvalue problems, Maxwell's equations, and the elasticity equations.The paper presents a new theoretical approach to the design and understanding of finite element discretizations for a wide variety of systems of partial differential equations (PDEs), known as finite element exterior calculus. This approach utilizes tools from differential geometry, algebraic topology, and homological algebra to develop discretizations that are compatible with the geometric, topological, and algebraic structures underlying well-posedness of PDE problems. The key idea is to view finite element spaces as spaces of piecewise polynomial differential forms, which connect to each other in discrete subcomplexes of elliptic differential complexes. These spaces are also related to the continuous elliptic complex through projections that commute with the complex differential. Applications of this approach include the finite element discretization of the Hodge Laplacian, Maxwell's equations, the equations of elasticity, and elliptic eigenvalue problems, as well as the construction of preconditioners. The paper also discusses the importance of stability in finite element methods, which is closely related to the well-posedness of the discrete equations. Stability is a subtle matter and can be very challenging to achieve, especially for important problems. The paper introduces the concept of finite element exterior calculus, which involves the development of finite element subcomplexes of certain elliptic differential complexes and cochain projections onto them. These subcomplexes inherit the cohomology and other features of the exact complexes. The main theme of the paper is the development of finite element subcomplexes of certain elliptic differential complexes and cochain projections onto them, and their implications and applications in numerical PDEs. The paper also discusses the computational challenges that motivated the development of finite element exterior calculus, including the poor behavior of seemingly reasonable numerical methods and the development of effective methods. The finite element exterior calculus provides an explanation for the difficulties experienced with naive methods and also points to a practical finite element solution. The paper presents a detailed discussion of the finite element exterior calculus, including the development of finite element spaces, the use of the Koszul complex, and the application of the Hodge star operation. It also discusses the use of group representation theory to characterize the spaces of polynomial differential forms used to construct finite element differential forms. The paper concludes with a discussion of the applications of finite element exterior calculus to various problems, including the Hodge Laplacian, eigenvalue problems, Maxwell's equations, and the elasticity equations.