Understanding heat at the atomistic level is an interesting exercises. It is fascinating to note how the vibration of atoms result into thermodynamic concept of heat. This thesis aims to bring insights into different constitutive laws of heat conduction. We also develop a framework in which the interaction of thermostats to the system can be studied and a well known Kapitza effect can be reduced. The thesis also explores stochastic and continuum methods to model the latent heat release in the first order transition of ideal silicon surfaces into dimers. We divide the thesis into three works which are connected to each other: 1. Fourier's law leads to a diffusive model of heat transfer in which a thermal signal propagates infinitely fast and the only material parameter is the thermal conductivity. In micro- and nano-scale systems, non-Fourier effects involving coupled diffusion and wavelike propagation of heat can become important. An extension of Fourier's law to account for such effects leads to a Jeffreys-type model for heat transfer with two relaxation times. In this thesis, we first propose a new "Thermal Parameter Identification" (TPI) method for obtaining the Jeffreys-type thermal parameters from molecular dynamics simulations. The TPI method makes use of a nonlinear regression-based approach for obtaining the coefficients in analytical expressions for cosine and sine-weighted averages of temperature and heat flux over the length of the system. The method is applied to argon nanobeams over a range of temperature and system sizes. The results for thermal conductivity are found to be in good agreement with standard Green-Kubo and direct method calculations. The TPI method is more efficient for systems with high diffusivity and has the advantage, that unlike the direct method, it is free from the influence of thermostats. In addition, the method provides the thermal relaxation times for argon. Using the determined parameters, the Jeffreys-type model is able to reproduce the molecular dynamics results for a short-duration heat pulse where wavelike propagation of heat is observed thereby confirming the existence of second sound in argon. Implementations of the TPI method in MATLAB are available as part of the supplementary material. 2. The second major work of the thesis is to look into the following problem. The direct method for computing thermal conductivity in nonequilibrium molecular dynamics gives rise to an artificial Kapitza resistance at the interface between thermostatted and unthermostatted regions. This resistance, which depends on the system size and the thermostat parameters, creates discontinuous jumps in the temperature and heat flux across the interface and therefore affects the measured thermal conductivity. In this part, we propose a phenomenological relation for the Kapitza resistance that can be used to extract a value for the bulk thermal conductivity, which is independent of the system size and thermostat details. We also provide insight into the Kapitza phenomenon resulting from numerical thermostatting. 3. This constitutes our third part of the thesis. Silicon (001) surfaces in non-equilibrium molecular dynamics (NEMD) simulations above a critical transformation temperature undergo a reconstruction from the ideal diamond crystalline surface to a reconstructed structure involving the formation of rows of dimers along a <110> direction. This process is accompanied by latent heat release that in NEMD simulations results in a dramatic increase in temperature of nanobeams that cross the transformation temperature as they are heated. To model this behavior, we propose a hybrid continuum partial differential equation for non-Fourier heat transfer coupled with a stochastic kinetic Monte Carlo (KMC) algorithm to account for latent heat release. An input to the method is the energy barrier for dimerization, which is computed separately using nudged elastic band calculations. The time-dependent temperature profiles along the beam predicted by the continuum-KMC method are in good agreement with the NEMD results.