Some years after the definition of braid groups by Artin, Zariski
introduced new topological groups, which were a natural extension of both
the classical braid group $B_n$ and the fundamental group of a surface.  
These groups then have been ``rediscovered" in the 60's in the study of
configuration spaces and they have been called \emph{surface braid
groups}.  Surface braid groups are related to mapping class groups,
classifying spaces and links in $3$-manifolds.  In the last years the
interest for these groups grew notably and several properties for braid
groups (and their singular extensions) have been generalised to surface
braid groups.  Several important results have been recently obtained on
braids; there are linear, orderable groups and the solution of word and
conjugation problem have been greatly improved and extended to Artin-Tits
and Garside groups, which are algebraic generalisations of braid groups.
On the other hand, in the case of surface braid groups there are many open
questions that should be answered:  Are these groups linear?  Are they
cohopfian?  How difficult is the conjugacy problem in these groups?  In
this talk we summarize some recent results on algebraical properties of
surface braid groups which might help to better understand the structure
of such groups and their relations with Artin-Tits groups and mapping
class groups.