Electronic-structure calculations based on hybrid functionals have emerged as a standard technique used in physics, chemistry, and material science. Despite this success, hybrid functionals have the drawback of containing undetermined parameters. To overcome this deficiency, two different nonempirical determination schemes are being investigated at present, namely dielectric-dependent hybrid (DDH) functionals and hybrid functionals that satisfy Koopmans' condition. This thesis is dedicated to the examination and further development of these approaches. In particular, we show that a precise description of band gaps in condensed-matter systems can be achieved with these functionals. Moreover, we demonstrate that their accuracy is comparable to state-of-the-art $GW$ methods while requiring substantially lower computational cost. First, we focus on the development of hybrid functionals satisfying Koopmans' condition. We show that the construction of those functionals can be optimized through suitably defined potential probes. By monitoring the delocalized screening charge, we achieve a measure of the degree of hybridization with the band states, which can be used to improve the band-gap estimate. We show that the application of this methodology to common semiconducting and insulating materials yields band gaps differing by less than 0.2 eV from experiment. These conceptual developments are an important step towards establishing hybrid functionals satisfying Koopmans' condition as a robust approach for band-gap predictions. Second, we examine DDH functionals and hybrid functionals satisfying Koopmans' condition for band gaps of more sophisticated materials. In particular, we focus on inorganic metal-halide perovskites which have recently drawn great scientific attention. For this class of materials, we show that both nonempirical hybrid-functional schemes yield band gaps of comparable accuracy ($\sim$0.2 eV) with respect to $GW$ reference calculations. Furthermore, we discuss the suitability of nonempirical hybrid-functional schemes for the application to the screening of large sets of perovskite materials. Third, we investigate the fundamental band gap of liquid water and hexagonal ice. These materials are particularly challenging since experimental studies have not reached a consensus on the band gap yet. Therefore, we first deduce robust benchmarks on the basis of a critical review of various experimental studies in the literature. Then, we compute the band gap through state-of-the-art $GW$ methods as well as nonempirical hybrid functionals. We show that theoretical calculations and experimental references are in good agreement with each other and we discuss critical aspects which are essential to ensure a consistent description of the band gap. Finally, we investigate band-edge levels as obtained with hybrid-functional calculations. The CaF$_2$/Si(111) interface serves thereby as an ideal test case to examine the accuracy of different theoretical schemes. The comparison with experiment reveals that global hybrid functionals and self-consistent $GW$ methods provide the most accurate description of the interfacial band alignment.