An infinite dimensional Lie algebra is locally finite if every finitely generated subalgebra is finite dimensional. On one extreme are the simple, locally finite Lie algebras. We provide structure theorems which describe such algebras over fields of positive characteristic. On the other extreme are the maximal, locally solvable Lie algebras, which are Borel subalgebras. We provide a theorem which shows that such Lie algebras are stabilizers of maximal, generalized flags, which is a generalization of Lie's theorem. We will finish by describing some new directions in the study of these Lie algebras.