  In this work some issues in the context of Noncommutative Quantum Mechanics
(NCQM) are addressed. The main focus is on finding whether symmetries present
in Quantum Mechanics still hold in the phase-space noncommutative version. In
particular, the issues related with gauge invariance of the electromagnetic
field and the weak equivalence principle (WEP) in the context of the
gravitational quantum well (GQW) are considered. The question of the Lorentz
symmetry and the associated dispersion relation is also examined. Constraints
are set on the relevant noncommutative parameters so that gauge invariance and
Lorentz invariance holds. In opposition, the WEP is verified to hold in the
noncommutative set up, and it is only possible to observe a violation through
an anisotropy of the noncommutative parameters.

General Relativity and Quantum Cosmology