We present the software design of Gridap, a novel finite element library written exclusively in the Julia programming language, which is being used by several research groups world-wide to simulate complex physical phenomena such as magnetohydrodynamics , photonics , weather modeling, non-linear solid mechanics, and fluid-structure interaction problems. The library provides a feature-rich set of discretization techniques for the numerical approximation of a wide range of mathematical models governed by Partial Differential Equations (PDEs), including linear, nonlinear, single-field, and multi-field equations. An expressive API allows users to define PDEs in weak form by a syntax close to the mathematical notation. While this is also available in previous frameworks, the main novelty of Gridap is that it implements this API without introducing a domain-specific language plus a compiler of variational forms. Instead, it leverages the Julia just-in-time compiler to build efficient code, specialized for the concrete problem at hand. As a result, there is no need to use different languages for the computational back-end and the user front-end anymore, thus eliminating the so-called two-language problem. Gridap also provides a low-level API that is modular and extensible via the multiple-dispatch paradigm of Julia and provides easy access to the main building blocks of the library if required. The main contribution of this paper is the detailed presentation of the novel software abstractions behind the Gridap design that leverages the new software possibilities provided by the Julia language. The second main contribution of the article is a performance comparison against FEniCS. We measure CPU times needed to assemble discrete systems of linear equations for different problem types and show that the performance of Gridap is comparable to FEniCS, demonstrating that the new software design does not compromise performance. Gridap is freely available at Github (github.com/gridap/Gridap.jl) and distributed under an MIT lice...