Understanding molecular symmetries has many important applications in chemistry. Symmetry is used in interpreting results in crystallography, spectroscopy, and quantum chemistry, as well as in classifying molecules. Mirror image symmetry is particularly important in predicting reactions and designing new pharmaceutical products. The group of rigid symmetries of a molecule known as the point group is widely used for classifying molecular symmetries. However, some molecules can rotate around particular bonds, and others are large enough to be somewhat flexible. Such molecules may have symmetries which are not included in the point group, and may even interconvert with their mirror image as a result of their flexibility. In this case, a topological approach to understanding molecular symmetries is more useful than a geometric one. This talk will present a survey of results about topological symmetries of molecular structures.