  This overview paper is intended as a quick introduction to Lie algebras of
vector fields. Originally introduced in the late 19th century by Sophus Lie to
capture symmetries of ordinary differential equations, these algebras, or
infinitesimal groups, are a recurring theme in 20th-century research on Lie
algebras. I will focus on so-called transitive or even primitive Lie algebras,
and explain their theory due to Lie, Morozov, Dynkin, Guillemin, Sternberg,
Blattner, and others. This paper gives just one, subjective overview of the
subject, without trying to be exhaustive.
