Nanotechnology has made it possible to create materials with unique properties. This development offers new opportunities and overcomes challenges for many thermal transport applications. Yet, it requires a more fundamental scientific understanding of nanoscale transport. This thesis emphasizes how simulation, mathematical, and numerical methods can lead to more grounded studies of nanoscale thermal transport for biological and physical applications. For instance, magnetic fluid hyperthermia (MFH), an emerging cancer treatment, is a noninvasive method to selectively destroy a tumor by heating a ferrofluid-impregnated malignant tissue with minimal damage to the surrounding healthy tissue. We model the problem by considering an idealized spherical tumor that is surrounded by healthy tissue. The dispersed magnetic nanoparticles in the tumor are excited by an AC magnetic field to generate heat. The temperature distribution during MFH is investigated through a bioheat transfer relation which indicates that the P\'eclet, Joule, and Fourier numbers are the more influential parameters that determine the heating during such a thermotherapy. Thus, we show that a fundamental parametric investigation of the heating of soft materials can provide pathways for optimal MFH design. Since ferrofluid materials themselves play a key role in heating, we examine six materials that are being considered as candidates for MFH use. These are simulated to investigate the heating of ferrofluid-loaded tumors. We show that iron-platinum, magnetite, and maghemite are viable MFH candidates since they are able to provide the desired heating of a tumor which will destroy it while keeping the surrounding healthy tissues at a relatively safe temperature. Recent advances in the synthesis and nanofabrication of electron devices have lead to diminishing feature sizes. This has in turn increased the power dissipation per unit area that is required to cool the devices, leading to a serious thermal management challenge. The phonon thermal conductivity is an important material property because of its role in thermal energy transport in semiconductors. A higher thermal conductivity material is capable of removing more heat since higher frequency phonons are able to travel through it. In this thesis, the effects of surface stress on the lattice thermal conductivity are presented for a silicon nanowire. Based on a continuum approach, a phonon dispersion relation is derived for a nanowire that is under surface stress and the phonon relaxation time is employed to subsequently determine its thermal conductivity. The surface stress is found to significantly influence the phonon dispersion and thus the Debye temperature. Consequently, the phonon thermal conductivity decreases with increasing surface stress. Different magnitudes of surface stress could arise from various material coatings and through different nanofabrication processes, effects of which are generally unclear and not considered. Our results show how such variations in surface stress can be gainfully used in phonon engineering and to manipulate the thermal conductivity of a nanomaterial. The thermal transport during thermoelectric cooling is also an important property since thermoelectric devices are compact, reliable, easy to control, use no refrigerants and require lower maintenance than do more traditional refrigeration devices. We focus on the Thomson effect that occurs when there is a current flow in the presence of a temperature gradient in the material, and investigate its influence on an intrinsic silicon nanowire cooler. The temperature dependence of the Thomson effect has a significant influence on the cooling temperature. We also consider thermal nonequilibrium between electrons and phonons over the carrier cooling length in the nanowire. The results show that a strong energy exchange between electrons and phonons lowers the cooling performance, suggesting useful strategies for thermoelectric device design. / Ph. D.