We study Ising ferrimagnets on square lattices with antiferromagnetic exchange couplings between spins of values S=1/2 and S=1 on neighbouring sites, couplings between S=1 spins at next--nearest neighbour sites of the lattice, and a single--site anisotropy term for the S=1 spins. Using mainly ground state considerations and extensive Monte Carlo simulations, we investigate various aspects of the phase diagram, including compensation points, critical properties, and temperature dependent anomalies. In contrast to previous belief, the next--nearest neighbour couplings, when being of antiferromagnetic type, may lead to compensation points.
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