In this paper, we consider the problem of quantization of classical Stackel systems and the problem of separability of related quantum Hamiltonians. First, using the concept of Stackel transform, natural Hamiltonian systems from a given Riemann space are expressed by some flat coordinates of related Euclidean configuration space. Then, the so-called flat minimal quantization procedure is applied in order to construct an appropriate Hermitian operator in the respective Hilbert space. Finally, we distinguish a class of Stackel systems which remains separable after any of admissible flat minimal quantizations. (C) 2014 Elsevier Inc. All rights reserved.