Skyrmions - A Theory of Nuclei surveys 60 years of research into the brilliant and imaginative idea of Tony Skyrme that atomic nuclei can be modelled as Skyrmions, topologically stable states in an effective quantum field theory of pions. Skyrme theory emerges as a low-energy approximation to the more fundamental theory of quarks and gluons - quantum chromodynamics (QCD). Skyrmions give spatial structure to the protons and neutrons inside nuclei, and capture the interactions of these basic particles, allowing them to partially merge. Skyrme theory also gives a topological explanation for the conservation of baryon number, a fundamental principle of physics. The book summarises the particle and field theory background, then presents Skyrme field theory together with the mathematics needed to understand it. Many beautiful and surprisingly symmetric Skyrmions are described and illustrated in colour. Quantized Skyrmion motion models the momentum, energy and spin of nuclei, and also their isospin, the quantum number distinguishing protons and neutrons. Skyrmion vibrations also need to be quantized, and the book reviews how the complicated energy spectra of several nuclei, including Carbon-12 and Oxygen-16, are accurately modelled by rotational/vibrational states of Skyrmions. A later chapter explores variants of Skyrme theory, incorporating mesons heavier than pions, and extending the basic theory to include particles like kaons that contain strange quarks. The final chapter introduces the Sakai-Sugimoto model, which relates Skyrmions to gauge theory instantons in a higher-dimensional framework inspired by string theory.