In this colloquium, I will focus on the statistical mechanics of chirality and biaxiality in liquid crystals through variety of theoretical approaches, including Monte Carlo simulations, Mean field calculation and Landau theory. All of these calculations for a model with dipole-induced-dipole interaction between biaxial molecules show that there is an important interaction between cholesteric twist and biaxiality. It is found that the cholesteric twist acts as a field on the biaxial order, and conversely, the biaxial order increases the twist i.e. reduces the pitch. This should be a promising area for experimental research, to further characterize the close relationship between chirality and biaxility.

In addition, I will present a brief overview of the studies on the following problems: phase transitions among different SmCP phases of bent-core liquid crystal induced by an electric field, splay flexoelectric effect in nematic phase of pear-shaped molecules and bend flexoelectric effect in bent-core nematic.