  We present covariant quantization rules for nonsingular finite dimensional
classical theories with flat and curved configuration spaces. In the beginning,
we construct a family of covariant quantizations in flat spaces and Cartesian
coordinates. This family is parametrized by a function $\omega(\theta)$,
$\theta\in\left( 1,0\right)$, which describes an ambiguity of the quantization.
We generalize this construction presenting covariant quantizations of theories
with flat configuration spaces but already with arbitrary curvilinear
coordinates. Then we construct a so-called minimal family of covariant
quantizations for theories with curved configuration spaces. This family of
quantizations is parametrized by the same function $\omega \left( \theta
\right)$. Finally, we describe a more wide family of covariant quantizations in
curved spaces. This family is already parametrized by two functions, the
previous one $\omega(\theta)$ and by an additional function $\Theta \left(
x,\xi \right)$. The above mentioned minimal family is a part at $\Theta =1$ of
the wide family of quantizations. We study constructed quantizations in detail,
proving their consistency and covariance. As a physical application, we
consider a quantization of a non-relativistic particle moving in a curved
space, discussing the problem of a quantum potential. Applying the covariant
quantizations in flat spaces to an old problem of constructing quantum
Hamiltonian in Polar coordinates, we directly obtain a correct result.
