There exist two correspondences between groups and Lie algebras. One occurs between matrix Lie groups and Lie algebras. The other concerns itself with complete groups of unitriangular matrices and Lie algebras. Titled the Lie and Mal'cev correspondences respectively, the purpose of this paper is to explore the two. We begin with an introduction to the basic properties of Lie algebras and other preliminary material followed by a construction of free Lie rings and algebras as well as by other interesting material discovered along the way. We then dive into the Lie correspondence, with which we contrast the Mal'cev correspondence in the section after.