Finite element exterior calculus (FEEC) is a framework for designing stable and accu- rate finite element discretizations for a wide variety of PDEs (e.g., Navier–Stokes equa- tions and Maxwell’s equations). The involved finite element spaces are constructed using piecewise polynomial differential forms, and stability of the discrete problems is estab- lished by preserving at the discrete level the geometric, topological, algebraic and analytic structures that ensure well-posedness of the continuous problem. In this talk, I will discuss the use of smooth splines within FEEC; they are the de facto standard for representing geometries of interest in engineering and offer superior accuracy in numerical simulations (per degree of freedom) compared to classical finite elements. I will discuss spline-based discretizations of the de Rham complex that go beyond the use of B-splines, focusing specifically on generalized B-splines and hierarchical B-splines. I will also present applications of the spline differential forms to the Navier–Stokes and Maxwell’s systems of equations.