Some of the most readily available and inexpensive monomers for ring-opening metathesis polymerization (ROMP) are norbornenes or substituted norbornadienes. Polymers made from them have tacticities (the stereochemical relationship between monomer units in the polymer chain) that remain after the C═C bonds in the polymer backbone are hydrogenated. Formation of polymers with exclusively a single structure (one tacticity) was rare until approximately 20 years ago, when well-defined ROMP catalysts based on molybdenum imido alkylidene complexes that contain a chiral biphenolate or binaphtholate ligand were shown to yield cis,isotactic-poly(2,3-dicarbomethoxynorbornadiene) and related polymers through addition of the monomer to the same side of the M═C bond in each step. Over the past few years, molybdenum and tungsten monoaryloxide pyrrolide (MAP) imido alkylidene initiators have been found to produce cis,syndiotactic polynorbornenes and substituted norbornadienes through addition of the monomer to one side of the M═C bond in one step followed by addition to the other side of the M═C bond in the next step. This "stereogenic metal control" is possible as a consequence of the fact that the configuration of the stereogenic metal center switches with each step in the polymerization. Stereogenic metal control also allows syndiotactic polymers to be prepared from racemic monomers in which enantiomers of the monomer are incorporated alternately into the main chain. Because pure trans polymers have not yet been prepared through some predictable mechanism of stereochemical control, it seems unlikely that all four basic polymer structures from a single given monomer can be prepared simply by choosing the right initiator. However, because tactic, and relatively oxygen-stable, hydrogenated polymers are often a desirable goal, the ability to form pure cis,isotactic polymers (through enantiomorphic site control) and cis,syndiotactic polymers (through stereogenic metal control) is sufficient for preparing hydrogenated polymers with a single structure. It is hoped that the principles of forming polymers that have a single structure through ring-opening metathesis polymerization will be general for a relatively large number of monomers and that some important problems in ROMP polymer chemistry can benefit from knowledge of polymer structure at a molecular level. With an increase in knowledge concerning the mechanistic details of polymerization by well-defined initiators, more elaborate ROMP polymers and copolymers with stereoregular structures may be possible.