  A systematic approach to the study of semiclassical fluctuations of strings
in AdS_5 x S^5 based on the Green-Schwarz formalism is developed. We show that
the string partition function is well defined and finite. Issues related to
different gauge choices are clarified. We consider explicitly several cases of
classical string solutions with the world surface ending on a line, on a circle
or on two lines on the boundary of AdS. The first example is a BPS object and
the partition function is one. In the third example the determinants we derive
should give the first corrections to the Wilson loop expectation value in the
strong coupling expansion of the n=4 SYM theory at large N.
