  We review the recent advances towards finding the spectrum of the AdS_5 x S^5
superstring. We thoroughly explain the theoretical techniques which should be
useful for the ultimate solution of the spectral problem. In certain cases our
exposition is original and cannot be found in the existing literature. The
present Part I deals with foundations of classical string theory in AdS_5 x
S^5, light-cone perturbative quantization and derivation of the exact
light-cone world-sheet scattering matrix.
