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Electronic structure calculations have become an indispensable tool in many areas of materials science and quantum chemistry. Even though the Kohn-Sham formulation of the density-functional theory (DFT) simplifies the many-body problem significantly, one is still confronted with several numerical challenges. In this article we describe how we have tackled these challenges by combining the projector augmented wave (PAW) method with uniform real-space grids, realized in the GPAW program package. Compared to more traditional plane wave or localized basis set approaches, real-space grids offer some advantages, most notably good parallelization possibilities and systematic convergence properties. However, as localized orbitals provide a conveniently small basis in some cases, we have also implemented atom-centered orbital basis sets in GPAW. While DFT allows one to study ground state properties, time-dependent density-functional theory provides access to the excited states. We have implemented the two common formulations of time-dependent density-functional theory, namely the linear response and the time propagation schemes. Electron transport calculations under finite-bias conditions can be performed with GPAW using non-equilibrium Green functions and the local basis set. In addition to the basic features of the real-space PAW method, we describe also the implementation of selected exchange-correlation functionals, the parallelization scheme as well as more special features such as the ΔSCF-method, calculation of X-ray absorption spectra, and maximally localized Wannier orbitals.