Finite groups of Lie type arise as the rational point over a finite field of a reductive linear algebraic group.
A standard technique to gain knowledge about representations of these groups and to classify them consist in detecting a suitable family of subgroups and building representations of the group by induction starting from the ones of the subgroups. The "classical" instance of this general idea Is the so called "Harish-Chandra theory", that is the study of representations by exploiting parabolic induction from Levi subgroups. Toward the end of last century, Deligne and Lusztig developed an enhancement of this theory, constructing a new induction that allows to keep track of "twisted" object.
My aim is to give an overview of some of the constructions involved and of the main results in these theories.