  This paper will appear in the Santa Cruz proceedings. An overview of the
braid group techniques in the theory of algebraic surfaces, from Zariski to the
latest results, is presented. An outline of the Van Kampen algorithm for
computing fundamental groups of complements of curves and the modification of
Moishezon-Teicher regarding branch curves of generic projections are given. The
paper also contains a description of a quotient of the braid group, namely
$\tilde B_n$ which plays an important role in the description of fundamental
groups of complements of branch curves.It turns out that all such groups are
``almost solvable'' $\tildeB_n$-groups. Finally, possible applications to the
study of moduli spaces of surfaces of general type are described and new
examples of positive signature spin surfaces whose fundamental groups can be
computed using the above algorithm (Galois cover of Hirzebruch surfaces) are
presented.
