  Non-associative finite invertible loops (NAFIL) are loops whose every element
has a unique two-sided inverse. Not much is known about the class of NAFIL
loops which includes the familiar IP (Inverse Property), Moufang, and Bol
loops. Our studies have shown that they are involved in such diverse fields as
combinatorics, finite geometries, quasigroups and related systems, Cayley
algebras, as well as in theoretical physics. This paper presents and
introductions to the class of NAFIL loops as the starting point for the
development of the theory of these interesting structures.
