Simulating thin structures implies solving non-linear partial differential equations defined on manifolds, which can include higher-order derivatives and/or stiff terms. Implementing effective locking-free discretisation techniques for nonlinear plates and shells is a challenging and highly technical task. We present FENICS-SHELLS, an open-source library for simulating thin structures using the finite element method, based on the FENICS PROJECT. Leveraging the high-level symbolic capabilities of the Unified Form Language (UFL) and the automatic code generation facilities of the FENICS Form Compiler (FFC), FENICS-SHELLS allows for the concise formulation of discretisations of various thin-structural theories, including linear and nonlinear geometrically exact Naghdi shell models, the Marguerre-von Kármán shallow shell model, and the Reissner-Mindlin plate model. Mixed Interpolation of Tensorial Component (MITC) and Partial Selective Reduced Integration (PSRI) numerical methods are used to alleviate shear-and membrane-locking issues. The effectiveness of these approaches and the ease of writing solvers for non-linear shell models is illustrated through a large set of verification tests and demo codes. FENICS-SHELLS can run without modification on high-performance computing (HPC) architectures.