Spurred by the development of new experimental techniques for designing and controlling exotic quantum many-body phenomena, the search for new theoretical concepts and computational algorithms has gained a lot of attention in physics research. Tensor networks has surfaced as a new language for describing quantum systems, because they allow for an explicit parametrization of the quantum correlations or entanglement as the driving force behind these phenomena. Taking place within the tensor-network endeavour, in this research proposal I intend to focus on describing and simulating quantum phases of two-dimensional systems with projected entangled-pair states (PEPS). I want to take this research in three directions, as I aim to (i) further develop PEPS tangent-space methods for simulating low-energy dynamics, (ii) explore the PEPS manifold for trial wavefunctions that describe exotic quantum phases of matter, and (iii) apply PEPS methods to open problems in condensed-matter physics. These three directions constitute a unified research program: not only will the development of more advanced methods allow for better simulations of real-life systems, and will a better insight into the structure and power of PEPS wavefunctions serve as a guide for interpreting simulation results, but keeping a focus on applications will also help in designing the crucial algorithms for the community and studying the interesting phases of matter.