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Recent advances in nonlinear photonics have enabled a new class of broadband ultra-stable light sources known as optical frequency combs. These light sources have given rise to an array of new optical devices and systems, spanning applications such as spectroscopy, astronomy, remote sensing, frequency synthesis, attoscience, telecommunications, and optical clockwork. At this time, there are a number of unsolved problems within the field. Optical frequency combs are often constrained to wavelengths within the near-infrared (NIR) due to the limited variety of in mature laser gain media and host glasses, and many applications such as spectroscopy, sensing, and attoscience would benefit from the development of optical frequency combs at longer wavelength ranges such as the mid-infrared (MIR). Furthermore, the generation and stabilization of frequency combs often requires rather complicated nonlinear optical systems, which have prevented these light sources from being used outside of dedicated optics labs. This dissertation considers new approaches to frequency comb generation based on recently discovered nonlinear dynamical processes that occur in quasi-phasematched (QPM) devices with quadratic nonlinearities. A recurring theme is that the interplay of nonlinear optical effects, such as optical parametric amplification and self-phase modulation, with linear optical effects, such as dispersion, can produce qualitatively new dynamical regimes. In many cases, these dynamical regimes exhibit favorable features that potentially solve the problems discussed above. The first half of this thesis considers the pulse formation mechanisms present in optical parametric oscillators (OPOs), and discusses new operating regimes that enable the generation of MIR combs with substantially more bandwidth than the NIR comb used to drive the OPO. These devices can produce few-cycle pulses with conversion efficiencies exceeding 50% while also preserving the coherence of the frequency comb. The latter portion of this thesis studies the dynamics of femtosecond pulses in nanophotonic waveguides. Here, the geometric dispersion associated with sub-wavelength confinement be used to achieve long interaction lengths with femtosecond pulses. Using these effects we are able to achieve saturated SHG with femtojoules of pulse energy, where state-of-the-art devices previously used picojoules. In the limit of phase-mismatched SHG driven with picojoules of pulse energy we observe the formation of a coherent multi-octave supercontinuum comprised of multiple spectrally broadened harmonics. The mechanisms of spectral broadening in this system are shown to be completely unique to dispersion-engineered nanophotonic QPM devices and exhibit a number of desirable features including i) low power requirements, ii) fewer decoherence mechanisms than traditional approaches, and iii) the formation of carrier-envelope-offset beatnotes in the regions of spectral overlap between the harmonics