Isotropy irreducible varieties are homogeneous complex
affine varieties whose tangent spaces are irreducible representations.
While geometry of symmetric isotropy irreducible varieties is a
classical topic, geometry of non-symmetric isotropy irreducible
varieties has not been well-understood. In this talk, first, I will
recall the notion of the isotropy irreducible varieties and their
classification, together with some interesting examples. Next, I will
present my recent result on a connection between isotropy irreducible
varieties and complex contact geometry. Namely, I will interpret a
non-symmetric isotropy irreducible variety as a parameter space of a
family of submanifolds of an adjoint variety, tangent to the contact
structure. Finally, I will give a classification of homogeneous
Legendrian subvarieties of adjoint varieties and a description of
their moduli spaces, which are either isotropy irreducible varieties
or irreducible Hermitian symmetric spaces.