Real theta-groups and orbit classification

We study a class of algebraic groups called theta-groups. Multiple symmetry groups of interest arise as theta-group; a special feature of them is that their 'orbits' can be classified despite the groups often being infinite. The original construction of theta-groups and its study were over the complex numbers. We describe Vinberg's work and provide an overview of the newer area of real theta-groups, which appear in areas such as black hole theory in physics. The main aims of our work are to provide a comprehensive overview of complex and real theta-groups.