This paper is concerned with some simple lattice models of the entanglement complexity of polymers in dilute solution, with special reference to biopolymers such as DNA. We review a number of rigorous results about the asymptotic behaviour of the knot probabihty, the entanglement complexity and the writhe of a lattice polygon (as a model of a ring polymer) and discuss Monte Carlo results for intermediate length polygons. In addition we discuss how this model can be augmented to include the effect of solvent quality and ionic strength. We also describe a lattice ribbon model which is able to capture the main properties of an oriented ribbon-like molecule (such as duplex DNA).