Report of the Detailed Calculation of the Effective Potential in   Spacetimes with $S^1\times R^d$ Topology and at Finite Temperature
  In this paper we review the calculations that are needed to obtain the bosonic and fermionic effective potential at finite temperature and volume (at one loop). The calculations at finite volume correspond to $S^1\times R^d$ topology. These calculations appear in the calculation of the Casimir energy and of the effective potential of extra dimensional theories. In the case of finite volume corrections we impose twisted boundary conditions and obtain semi-analytic results. We mainly focus in the details and validity of the results. The zeta function regularization method is used to regularize the infinite summations. Also the dimensional regularization method is used in order to renormalize the UV singularities of the integrations over momentum space. The approximations and expansions are carried out within the perturbative limits. After the end of each section we briefly present applications associated to the calculations. Particularly the calculation of the effective potential at finite temperature for the Standard Model fields, the effective potential for warped and large extra dimensions and the topological mass creation. In the end we discuss on the convergence and validity of one of the obtained semi-analytic results. 
V.K.Oikonomou