  We measure the second-harmonic (SH) response of MoS$_2$ in a broad
fundamental photon energy range between 0.85 and 1.7 eV, and present a
continuous SH spectrum capturing signatures of both the fundamental A and B
excitons in addition to the intense C resonance. Moreover, we interpret our
results in terms of the first exciton simulation of the SH properties of
multi-layered MoS$_2$ samples, represented by a study of trilayer MoS$_2$. The
good agreement between theory and experiments allows us to establish a
connection between the measured spectrum and the underlying electronic
structure, in the process elucidating the non-linear excitation of
electron-hole pairs.
