  In this paper, we study normalized solutions to the
Chern-Simons-Schr\"odinger system, which is a gauge-covariant nonlinear
Sch\"oridnger system with a long-range electromagnetic field, arising in
nonrelativistic quantum mechanics theory. The solutions correspond to critical
points of the underlying energy functional subject to the $L^2$-norm
constraint. Our research covers several aspects. Firstly, in the mass
subcritical case, we establish the compactness of any minimizing sequence to
the associated global minimization problem. As a by-product of the compactness
of any minimizing sequence, the orbital stability of the set of minimizers to
the problem is achieved. In addition, we discuss the radial symmetry and
uniqueness of minimizer to the problem. Secondly, in the mass critical case, we
investigate the existence and nonexistence of normalized solution. Finally, in
the mass supercritical case, we prove the existence of ground state and
infinitely many radially symmetric solutions. Moreover, the instability of
ground states is explored
