The exponential map plays an important role in Lie theory; it allows one to linearize a Lie group in the neighborhood of the identity element, thus reducing many questions about Lie groups to (more tractable) questions about Lie algebras. Unfortunately (at least for an algebraic geometer), the exponential map is not algebraic; it is given by an infinite series and thus cannot be defined in the setting of algebraic groups. \vskip.2in \noindent The next best thing is to linearize the conjugation action of $G$ on itself in a Zariski neighborhood of the identity element. For special orthogonal groups $SO_n$ this is done by the classical Cayley map, which has been used in place of the exponential map in some applications. In the 1980s D. Luna asked which other simple algebraic groups admit a ``Cayley map". In this talk, I will discuss the background of this problem and a recent solution, obtained jointly with N. Lemire and V. L. Popov.