  The formulation of noncommutative quantum mechanics as a quantum system
represented in the space of Hilbert-Schmidt operators is used to systematically
derive, using the standard time slicing procedure, the path integral action for
a particle moving in the noncommutative plane and in the presence of a magnetic
field and an arbitrary potential. Using this action, the equation of motion and
the ground state energy for the partcle are obtained explicitly. The
Aharonov-Bohm phase is derived using a variety of methods and several dualities
between this system and other commutative and noncommutative systems are
demonstrated. Finally, the equivalence of the path integral formulation with
the noncommutative Schr\"{o}dinger equation is also established.
