The pelaron problem has been of interest in condensed matter physics cind held theory tor cibout half a century. Within the framework of Vcist variety of theoreticcil approximations, the bulk polaron properties have been extensively (explored and fairly well understood in the literature. In the last two deccides, with the impressive progress achieved in the mici-ofabrication technology, it became possible to ol)t£iin low dimensional microstructures, in which the charge ca.rriers are confined in one or more spatial dii'ections. Consequently, there has appeared (|uite a large interest in phonon coupling-induced effects and polaronic properties of low dimensionally confined electrons. In this context, this thesis work is devoted to the study of low dimensional optical polaron properties, with the application of several different formal approaches common in the literature, such as perturbation theory, variatioiicil principles and Feynman path integral formalism. The model we adopt in this work consists of an electron, confined within an external potential (quantuni well), and interacting via the Fröhlich Harniltonian with the bulk LO-phonons of the relevant well material. Therefore, our primary concern is to give a clear view of only the bulk phonon effects on an electron in confined media, and we disregard all other complications that may come about from screening effects, phonon confinement, etc. Under these assumptions, we calculate the ground state energy, the effective mass, and some other quantities of polaron in several confinement geometries. We also provide a broad interpolating overview to the one polaxon problem in the overall range of electron-phonon coupling constant and in a general type of confinement, which can be conformed from one geometriccd configuration to another. Another interesting theme of the polaron theory, magneto-polaron, is considered in the context of the confinement effect on the polaron, brought about by the rncignetic field. A detailed analysis is given in the case, where the effect of electron-phonon coupling is dominated over by the magnetic field counterpart of the problem.