  In this paper we study nonlinear interpolation problems for interpolation and
peak-interpolation sets of function algebras. The subject goes back to the
classical Rudin-Carleson interpolation theorem. In particular, we prove the
following nonlinear version of this theorem:
  Let $\bar{\mathbb D}\subset \mathbb C$ be the closed unit disk, $\mathbb
T\subset\bar{\mathbb D}$ the unit circle, $S\subset\mathbb T$ a closed subset
of Lebesgue measure zero and $M$ a connected complex manifold.
  Then for every continuous $M$-valued map $f$ on $S$ there exists a continuous
$M$-valued map $g$ on $\bar{\mathbb D}$ holomorphic on its interior such that
$g|_S=f$. We also consider similar interpolation problems for continuous maps
  $f: S\rightarrow\bar M$, where $\bar M$ is a complex manifold with boundary
$\partial M$ and interior $M$. Assuming that $f(S)\cap\partial M\ne\emptyset$
we are looking for holomorphic extensions $g$ of $f$ such that $g(\bar{\mathbb
D}\setminus S)\subset M$.
