  We give a survey of the theory of surface braid groups and the lower
algebraic K-theory of their group rings. We recall several definitions and
describe various properties of surface braid groups, such as the existence of
torsion, orderability, linearity, and their relation both with mapping class
groups and with the homotopy groups of the 2-sphere. The braid groups of the
2-sphere and the real projective plane are of particular interest because they
possess elements of finite order, and we discuss in detail their torsion and
the classification of their finite and virtually cyclic subgroups. Finally, we
outline the methods used to study the lower algebraic K-theory of the group
rings of surface braid groups, highlighting recent results concerning the braid
groups of the 2-sphere and the real projective plane.
