The work carried out in this thesis has been motivated by the promising applicability of photonic nanostructures in optical communications, internet data centers (ICD) and biosensing, to name a few. In particular, the dispersion and nonlinear engineering that silicon photonic crystal waveguides (Si-PhCWGs) and diamond-fin waveguides allow, can be exploited in the design of important photonic components, such as frequency comb generators, Raman amplifiers or filters. Within such objectives, we present rigorous and comprehensive theoretical models where all relevant linear and nonlinear optical effects, including modal dispersion, waveguide loss, free-carrier (FC), Kerr and Raman effects are considered. In the case of the newly developed subwavelength diamond-fin waveg- uides, we complete a detailed characterization of their dispersion and nonlinear optical properties, along with an analysis of pulsed dynamics in these structures. As a relevant application, we demonstrate how these waveguides can be employed to efficiently gener- ate soliton frequency combs in the visible spectral domain. With regards to Si-PhCWGs, we firstly explore the effect of stimulated Raman scattering in the slow-light regime, and demonstrate that signal amplification without pulse distortion can be achieved. Secondly, we add photonic crystal cavities (PhCCs) alongside the Si-PhCWG, with the associated inter-cavity coupling and waveguide-cavity interactions. Therefore, we describe a novel mathematical model and its corresponding computational tool that solves the dynamics of the forwards and backwards propagating pulses, the energy in the cavities and the FCs at the waveguide and at the cavities. Finally, we show the potential practical use of the model by simulating a photonic drop-filter with back reflection nulling.