  A constituent quark model, which has recently been successfully applied to
the study of heavy quarkonium properties such as its spectrum but also a
diverse array of observables related with their electromagnetic, strong and
weak decays and reactions, is used herein to compute ground- and excited-state
masses of $QQQ$-baryons containing either $c$- or $b$-quarks. Considering the
lack of experimental information about the spectra of triply-heavy baryons, we
believe that our computation could help on finding new states, since it is
expected that phenomenological quark models describe triply-heavy baryons to a
similar degree of accuracy as heavy quarkonia. The quark model parameters
previously used to describe $c\bar c$ and $b\bar b$ properties have not been
modified for this analysis. The non-relativistic three-body bound-state problem
is solved by means of the Gau\ss ian expansion method which provides enough
accuracy and simplifies the subsequent evaluation of the matrix elements.
Several low-lying states with quantum numbers $J^P=\frac{1}{2}^{\pm}$,
$\frac{3}{2}^{\pm}$, $\frac{5}{2}^{\pm}$ and $\frac{7}{2}^+$ are reported. We
compare our results with those predicted by many other theoretical formalisms.
There is a general trend of agreement about the mass of the ground state in
each sector of triply-heavy baryons; however, the situation is more puzzling
for the excited states and thus appropriate comments on the most relevant
features of our comparison are given.
