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.