  We present a rigorous derivation of a real space Full-Potential
Multiple-Scattering-Theory (FP-MST), valid both for continuum and bound states,
that is free from the drawbacks that up to now have impaired its development,
in particular the need to use cell shape functions and rectangular matrices. In
this connection we give a new scheme to generate local basis functions for the
truncated potential cells that is simple, fast, efficient, valid for any shape
of the cell and reduces to the minimum the number of spherical harmonics in the
expansion of the scattering wave function. The method also avoids the need for
saturating 'internal sums' due to the re-expansion of the spherical Hankel
functions around another point in space (usually another cell center). Thus
this approach, provides a straightforward extension of MST in the Muffin-Tin
(MT) approximation, with only one truncation parameter given by the classical
relation $l_{\rm max} = kR_b$, where $k$ is the excited (or ground state)
electron wave vector and $R_b$ the radius of the bounding sphere of the
scattering cell. It is shown that the theory converges absolutely in the
$l_{\rm max} \to \infty$ limit. As a consequence it provides a firm ground to
the use of FP-MST as a viable method for electronic structure calculations and
makes possible the computation of x-ray spectroscopies, notably photo-electron
diffraction, absorption and anomalous scattering among others, with the ease
and versatility of the corresponding MT theory. Some numerical applications of
the theory are presented, both for continuum and bound states.
