I will report on a joint work in progress with Hiroshi Ando where we follow and extend the recent results of Leonel Robert on the Lie group structure of the unitary group of a unital C*-algebra. I will give some introduction to infinite-dimensional linear Lie groups and describe e.g. how the ideal structure of the algebra corresponds to the normal subgroup structure of the unitary group, how the properties such as simplicity or unique trace property are visible from the structure of the corresponding unitary group, how to characterize simplicity of reduced group C*-algebras Lie-theoretically, etc.