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Accurate numerical simulation of reaction-diffusion systems can come with a high cost. A system may be stiff, and solutions may exhibit sharp localized features that require fine grids and small time steps to properly resolve the physical phenomena they represent. The development of efficient methods is crucial to cut down the demands of computational resources. In this thesis we consider the use of adaptive space and time methods driven by a posteriori error estimation. The error estimators for the spatial discretization are built from a variety of sources: the residual of the partial differential equation (PDE) system, gradient recovery operators and interpolation estimates. The interpolation estimates are anisotropic, not relying on classical mesh regularity assumptions. The adapted mesh is therefore allowed to include elements elongated in specified directions, as dictated by the type of solution being approximated. This thesis proposes an element-based adaptation method to be used for a residual estimator. This method avoids the usual conversion of the estimator to a metric, and instead applies the estimator to directly control the local mesh modifications. We derive a new error estimator for the L^2-norm in the same anisotropic setting and adjust the element-based adaptation algorithm to the new estimator. This thesis considers two new adaptive finite element settings for reaction-diffusion problems. The first is the extension to a PDE setting of an estimator for the time discretization with the backward difference formula of order 2 (BDF2), based on an estimator for ordinary differential equation (ODE) problems. Coupled with the residual estimator, we apply a space-time adaptation method. The second is the derivation of anisotropic error estimates for the monodomain model from cardiac electrophysiology. This model couples a nonlinear parabolic PDE with an ODE and this setting presents challenges theoretically as well as numerically. In addition to theoretical considerations, numerical tests are performed throughout to assess the reliability and efficiency of the proposed error estimators and numerical methods.