This paper gives a construction of braid group actions on the
derived category of coherent sheaves on a variety $X$. The motivation
for this is M. Kontsevich's homological mirror conjecture, together
with the occurrence of certain braid group actions in symplectic
geometry. One of the main results is that when dim $X\geq 2$, our
braid group actions are always faithful.
¶ We describe conjectural mirror symmetries between smoothings and
resolutions of singularities which lead us to find examples of braid
group actions arising from crepant resolutions of various
singularities. Relations with the McKay correspondence and with
exceptional sheaves on Fano manifolds are given. Moreover, the case of
an elliptic curve is worked out in some detail.