DOLFINx is the next generation problem solving environment from the FEniCS Project; it provides an expressive and performant environment for solving partial differential equations using the finite element method. We present the modern design principles that underpin the DOLFINx library, and describe approaches used in DOLFINx that preserve the high level of mathematical abstraction associated with FEniCS Project libraries, yet support extensibility and specialized customization. At the core of DOLFINx is a data- and function-oriented design, in contrast with the object-oriented design of more traditional libraries. We argue that this novel design approach leads to a compact and maintainable library, which is flexible in use and makes possible the creation of high performance programs in different languages.