  In this review paper we present a detailed account of the extraction and the
calculation of the total reaction cross section of strongly bound and weakly
bound, stable and unstable, exotic, nuclei. We discuss the optical model and
the more general coupled channels model of direct reactions. The effect of
long-range absorption due to the coupling to excited states in the target and
to the breakup continuum in the projectile is also discussed. The generalized
optical theorem for charged particle scattering and the resulting sum-of
differences method is then discussed. The so-called "quarter-point recipe" is
discussed next, and the quarter-point angle is introduced as a simple and rapid
mean to obtain the total reaction cross section. The last topic discussed is
the reduction of the total reaction cross section that would allow a large body
of data to sit on a single universal function. Such a universal function exists
in the case of the fusion data, and the aim of this last topic of the review is
to extend the fusion case to the total reaction, by adding the direct reaction
contribution. Also discussed is the inclusive breakup cross section and how it
can be used to extract the total reaction cross section of the interacting
fragment with the target. This method is also known as the Surrogate method and
represents a case of hybrid reactions. The sum of the integrated inclusive
breakup cross section with the complete fusion cross section supplies the total
fusion cross section. The main experimental methods to determine the total
reaction cross section are also discussed, with emphasis in recent techniques
developed to deal with reactions induced by unstable beams.
