  We recall some of the fundamental achievements of formal deformation
quantization to argue that one of the most important remaining problems is the
question of convergence. Here we discuss different approaches found in the
literature so far. The recent developments of finding convergence conditions
are then outlined in three basic examples: the Weyl star product for constant
Poisson structures, the Gutt star product for linear Poisson structures, and
the Wick type star product on the Poincar\'e disc.
