  Excitons in thin layers of semiconducting transition metal dichalcogenides
are highly subject to the strongly modified Coulomb electron-hole interaction
in these materials. Therefore, they do not follow the model system of a
two-dimensional hydrogen atom. We investigate experimentally and theoretically
excitonic properties in both the monolayer (ML) and the bilayer (BL) of
MoSe$_2$ encapsulated in hexagonal BN. The measured magnetic field evolutions
of the reflectance contrast spectra of the MoSe$_2$ ML and BL allow us to
determine $g$-factors of intralayer A and B excitons, as well as the $g$-factor
of the interlayer exciton. We explain the dependence of $g$-factors on the
number of layers and excitation state using first principles calculations.
Furthermore, we demonstrate that the experimentally measured ladder of
excitonic $s$ states in the ML can be reproduced using the $\mathbf{k\cdot p}$
approach with the Rytova-Keldysh potential that describes the electron-hole
interaction. In contrast, the analogous calculation for the BL case requires
taking into account the out-of-plane dielectric response of the MoSe$_2$ BL.
