  We propose a definition of ``nonabelian mixed Hodge structure'' together with
a construction associating to a smooth projective variety $X$ and to a
nonabelian mixed Hodge structure $V$, the ``nonabelian cohomology of $X$ with
coefficients in $V$'' which is a (pre-)nonabelian mixed Hodge structure denoted
$H=Hom(X_M, V)$. We describe the basic definitions and then give some
conjectures saying what is supposed to happen. At the end we compute an
example: the case where $V$ has underlying homotopy type the complexified
2-sphere, and mixed Hodge structure coming from its identification with $\pp
^1$. For this example we show that $Hom (X_M,V)$ is a namhs for any smooth
projective variety $X$.
