  We study sub-Riemannian and sub-Lorentzian geometry on the Lie group
$\SU(1,1)$ and on its universal cover $\CSU(1,1)$. In the sub-Riemannian case
we find the distance function and completely describe sub-Riemannian geodesics
on both $\SU(1,1)$ and $\CSU(1,1)$, connecting two fixed points. In particular,
we prove that there is a strong connection between the conjugate loci and the
number of geodesics. In the sub-Lorentzian case, we describe the geodesics
connecting two points on $\CSU(1,1)$, and compare them with Lorentzian ones. It
turns out that the reachable sets for Lorentzian and sub-Lorentzian normal
geodesics intersect but are not included one to the other. A description of the
timelike future is obtained and compared in the Lorentzian and sub-Lorentzain
cases.
