  In this manuscript we review recent developments in the numerical simulations
of bipartite SU(N) spin models by quantum Monte Carlo (QMC) methods. We provide
an account of a large family of newly discovered sign-problem free spin models
which can be simulated in their ground states on large lattices, containing
O(10^5) spins, using the stochastic series expansion method with efficient loop
algorithms. One of the most important applications so far of these Hamiltonians
are to unbiased studies of quantum criticality between Neel and valence bond
phases in two dimensions -- a summary of this body of work is provided. The
article concludes with an overview of the current status of and outlook for
future studies of the "designer" Hamiltonians.
