  We study the world-volume theory of a bosonic membrane perturbatively and
discuss if one can obtain any conditions on the number of space-time dimensions
from the consistency of the theory. We construct an action which is suitable
for such a study. In order to study the theory perturbatively we should specify
a classical background around which perturbative expansion is defined. We will
discuss the conditions which such a background should satisfy to deduce the
critical dimension. Unfortunately we do not know any background satisfying such
conditions. In order to get indirect evidences for the critical dimension of
the membrane, we next consider two string models obtained via double
dimensional reduction of the membrane. The first one reduces to the Polyakov
string theory in the conformal gauge. The second one is described by the Schild
action. We show that the critical dimension is 26 for these string theories,
which implies that the critical dimension is 27 for the membrane theory.
