This thesis concerns various theoretical proposals for engineering dispersion relations of photons and electrons for particular applications. The common concept is the use of a periodic modulation to induce new phenomena on length scales comparable with the periodicity of the modulation. In particular, the modulation leads to the emergence of band gaps, which are accompanied by a strongly modified density of states near and within the band gap. The main focus is on two applications of such modified densities of states. Firstly, the intentional introduction of defects in an otherwise perfectly periodic modulation of an electron gas leads to the emergence of localized defect states with energies within the band gap, where no propagating modes exist. Secondly, the divergence of the photonic density of states near a photonic band gap leads to strongly modified light-matter interactions, which has applications both in terms of spontaneous emission control and for slow light propagation. We first consider antidot lattices, periodic modulations of the potential of an electron gas. We demonstrate that such structures may serve as an interesting platform for quantum information processing. In particular, we discuss the use of the spins of electrons localized within defect states as spin qubits. We demonstrate, using numerically exact calculations, that coupling of such spin qubits via the exchange interaction can be efficiently tuned via electrostatic gates or external magnetic fields. By benchmarking with the numerically exact results we discuss the validity of certain approximative methods for calculating the exchange interaction. These ideas are applied also to graphene, where the emergence of a band gap is in itself interesting in that it turns the otherwise semimetallic graphene into a semiconductor, paving the way for graphene transistors. Photonic band gaps can be engineered using structures with a periodic modulations of the refractive index, commonly referred to as photonic crystal. We discuss the application of photonic crystals to slow light phenomena, where advantage is taken of the divergence of the density of states near the band gap edge. Using a perturbative approach, we demonstrate certain limits of the attainable slow down factors due to broadening of electromagnetic modes. We discuss the effect of damping due to a finite conductivity as well as structural disorder, and provide a common framework for including a wide range of broadening mechanisms.