We review the application of field-theoretic renormalization group (RG) methods to the study of fluctuations in reaction-diffusion problems. We first investigate the physical origin of universality in these systems, before comparing RG methods to other available analytic techniques, including exact solutions and Smoluchowski-type approximations. Starting from the microscopic reaction-diffusion master equation, we then pedagogically detail the mapping to a field theory for the single-species reaction kA --> ellA(ell < k). We employ this particularly simple but non-trivial system to introduce the field-theoretic RG tools, including the diagrammatic perturbation expansion, renormalization and Callan-Symanzik RG flow equation. We demonstrate how these techniques permit the calculation of universal quantities such as density decay exponents and amplitudes via perturbative epsi = dc - d expansions with respect to the upper critical dimension dc. With these basics established, we then provide an overview of more sophisticated applications to multiple species reactions, disorder effects, Lévy flights, persistence problems and the influence of spatial boundaries. We also analyse field-theoretic approaches to non-equilibrium phase transitions separating active from absorbing states. We focus particularly on the generic directed percolation universality class, as well as on the most prominent exception to this class: even-offspring branching and annihilating random walks. Finally, we summarize the state of the field and present our perspective on outstanding problems for the future.