We give an overview of studies of models for semiflexible, equilibrium polymers with special emphasis on our work on both lattice and continuum models for such systems. We show, principally by Monte Carlo simulations, that, once monomers self assemble to form polymers, their semiflexibility leads to nematic phases at low temperatures. Attractive wall potentials encourage the adsorption of these equilibrium polymers on surfaces. Rapid cooling leads to the formation of glasses with entangled polymers. Shear promotes nematic ordering, but, at high shear rates, this tendency decreases since the equilibrium polymers are torn apart. A version of our model in which the polymers are directed shows the polymer analog of bosonic Mott-insulating, mass-density-wave, and supersolid phases. We give a brief comparison of our work with other studies and also explore the experimental implications of our study.