** Abstract: **
The classical Cayley map,
X --> (I_n-X)/(I_n+X)
is a birational isomorphism between the
special orthogonal group SO_n and
its Lie algebra
so_n, which is SO_n-equivariant
with respect to the conjugating and
adjoint actions respectively.
We ask whether or not maps with these
properties can be constructed for other
algebraic groups.
We show that the answer is usually
``no", with a few exceptions. In
particular, we show that a Cayley map
for the group SL_n exists if and only
if n <= 3. This answers an old question
of Luna.