  We investigate Ising ferrimagnets on square and simple-cubic lattices with
exchange couplings between spins of values S=1/2 and S=1 on neighbouring sites
and an additional single-site anisotropy term on the S=1 sites. Based mainly on
a careful and comprehensive Monte Carlo study, we conclude that there is no
tricritical point in the two--dimensional case, in contradiction to mean-field
predictions and recent series results. However, evidence for a tricritical
point is found in the three-dimensional case. In addition, a line of
compensation points is found for the simple-cubic, but not for the square
lattice.
