We first discuss the construction by Perez-Izquierdo and Shestakov of
universal nonassociative enveloping algebras of Malcev algebras. We then
describe recent results on explicit structure constants for the universal
enveloping algebras (both nonassociative and alternative) of the 4-dimensional
solvable Malcev algebra and the 5-dimensional nilpotent Malcev algebra. We
include a proof (due to Shestakov) that the universal alternative enveloping
algebra of the real 7-dimensional simple Malcev algebra is isomorphic to the
8-dimensional division algebra of real octonions.