  A metric space $X$ has {\em Markov type} 2, if for any reversible
finite-state Markov chain $\{Z_t\}$ (with $Z_0$ chosen according to the
stationary distribution) and any map $f$ from the state space to $X$, the
distance $D_t$ from $f(Z_0)$ to $f(Z_t)$ satisfies $\E(D_t^2) \le K^2 t
\E(D_1^2)$ for some $K=K(X)<\infty$. This notion is due to K. Ball (1992), who
showed its importance for the Lipschitz extension problem. However until now,
only Hilbert space (and its bi-Lipschitz equivalents) were known to have Markov
type 2. We show that every Banach space with modulus of smoothness of power
type 2 (in particular, $L_p$ for $p>2$) has Markov type 2; this proves a
conjecture of Ball. We also show that trees, hyperbolic groups and simply
connected Riemannian manifolds of pinched negative curvature have Markov type
2. Our results are applied to settle several conjectures on Lipschitz
extensions and embeddings. In particular, we answer a question posed by Johnson
and Lindenstrauss in 1982, by showing that for $1<q<2<p<\infty$, any Lipschitz
mapping from a subset of $L_p$ to $L_q$ has a Lipschitz extension defined on
all of $L_p$.
