  Existence and uniqueness of complex geodesics joining two points of a convex
bounded domain in a Banach space $X$ are considered. Existence is proved for
the unit ball of $X$ under the assumption that $X$ is 1-complemented in its
double dual. Another existence result for taut domains is also proved.
Uniqueness is proved for strictly convex bounded domains in spaces with the
analytic Radon-Nikodym property. If the unit ball of $X$ has a modulus of
complex uniform convexity with power type decay at 0, then all complex
geodesics in the unit ball satisfy a Lipschitz condition. The results are
applied to classical Banach spaces and to give a formula describing all complex
geodesics in the unit ball of the sequence spaces $\ell^p$ ($1 \leq p <
\infty$).
