In this work we present a framework for the calculation of the conduction properties of a metal-molecule-metal junction which is in contact with its thermal environment. The effects of thermal relaxation and dephasing on the transmission properties of the junction were studied using a simple tight binding model for the molecular conductor. The interaction between the molecular system and the thermal environment is described on the level of the Redfield theory, which is a weak electron-phonon coupling scheme, modified for the description of steady-state situations. We show that the transmitted flux consists of two (generally non-separable) components: a flux associated with the elastic tunneling and a thermally activated flux component. The coherent (tunneling) component dominates the transport at low temperatures, large energy gaps and short molecular chains. The incoherent (activated) component is important in the opposite limits. The integrated transmission provides a generalization of the Landauer conduction formula in the presence of thermal relaxation. [1] Using the same formalism, we investigate the issue of heat release on a current carrying molecule: the total amount of heat that is generated on the wire during electrical conduction. Local aspects of the heat release were investigated as well [2]. We compare quantum to classical calculations in the resonance regime, and far from it. For the quantum case we calculate the fraction of the available energy, i.e. of the potential drop, that is converted to heat on the molecular barrier and its dependence on the system parameters: the potential bias, molecule length, dephasing rate and temperature. We find that in the localization limit, where the electron is fully thermalized at each molecular site, all the available energy is dissipated on the bridge, while for short systems and weak system-bath coupling, only a small fraction of the available energy is deposited as heat on the bridge. In our simulations we got this fraction to be of the order 0.1-0.3 using a reasonable range of parameters. We also present a scheme for the analysis of local aspects of heat release, and use it to study the position dependence of the power dissipation for a specific molecular structure. Finally, using classical heat conduction theory we estimate the temperature rise on the molecule due to the heating effects. We find that, within a reasonable range of voltage and molecular parameters, it is in the few degrees range, and therefore it should not affect the molecular junction functionality. It should be emphasized that classical heat transfer theory overestimates heat conduction, so a quantum treatment of vibrational energy transmission in molecular junctions is needed in order to better estimate this temperature rise of the molecular junction. Such study is currently underway.