  We consider Moebius and conformal homeomorphisms $f : \partial X \to \partial
Y$ between boundaries of CAT(-1) spaces $X,Y$ equipped with visual metrics. A
conformal map $f$ induces a topological conjugacy of the geodesic flows of $X$
and $Y$, which is flip-equivariant if $f$ is Moebius. We define a function
$S(f) : \partial ^2 X \to \mathbb{R}$, the {\it integrated Schwarzian} of $f$,
which measures the deviation of the topological conjugacy from being
flip-equivariant, in particular vanishing if $f$ is Moebius. Conversely if
$X,Y$ are simply connected complete manifolds with pinched negative sectional
curvatures, then $f$ is Moebius on any open set $U \subset \partial X$ such
that $S(f)$ vanishes on $\partial^2 U$. Indeed we obtain an explicit formula
for the cross-ratio distortion in terms of the integrated Schwarzian. For such
manifolds, we show that there is a Moebius homeomorphism $f : \partial X \to
\partial Y$ if and only if there is a topological conjugacy of geodesic flows
$\phi : T^1 X \to T^1 Y$ with a certain uniform continuity property along
geodesics. We show that if $X,Y$ are proper, geodesically complete CAT(-1)
spaces then any Moebius homeomorphism $f$ extends to a $(1, \log
2)$-quasi-isometry with image $\frac{1}{2}\log 2$-dense in $Y$. We prove that
if $X,Y$ are in addition metric trees then $f$ extends to a surjective
isometry. For $C^1$ conformal maps $f : \partial X \to \partial Y$ with bounded
integrated Schwarzian and with domain $X$ a simply connected negatively curved
manifold with a lower bound on sectional curvature, similar arguments show that
$f$ extends to a $(1, \log 2 + 12||S(f)||_{\infty})$ quasi-isometry. We also
obtain a dynamical classification of Moebius self-maps $f : \partial X \to
\partial X$ into three types, elliptic, parabolic and hyperbolic.
