  A primitive multiple scheme is a Cohen-Macaulay scheme $Y$ such that the
associated reduced scheme $X=Y_{red}$ is smooth, irreducible, and that $Y$ can
be locally embedded in a smooth variety of dimension $\dim(X)+1$. If $n$ is the
multiplicity of $Y$, there is a canonical filtration $X=X_1\subset
X_2\subset\cdots\subset X_n=Y$, such that $X_i$ is a primitive multiple scheme
of multiplicity $i$. The simplest example is the trivial primitive multiple
scheme of multiplicity $n$ associated to a line bundle $L$ on $X$: it is the
$n$-th infinitesimal neighborhood of $X$, embedded in the line bundle $L^*$ by
the zero section.
  The main subject of this paper is the construction and properties of fine
moduli spaces of vector bundles on primitive multiple schemes. Suppose that
$Y=X_n$ is of multiplicity $n$, and can be extended to $X_{n+1}$ of
multiplicity $n+1$, and let $M_n$ a fine moduli space of vector bundles on
$X_n$. With suitable hypotheses, we construct a fine moduli space $M_{n+1}$ for
the vector bundles on $X_{n+1}$ whose restriction to $X_n$ belongs to $M_n$. It
is an affine bundle over the subvariety $N_n\subset M_n$ of bundles that can be
extended to $X_{n+1}$. In general this affine bundle is not banal. This applies
in particular to Picard groups.
  We give also many new examples of primitive multiple schemes $Y$ such that
the dualizing sheaf $\omega_Y$ is trivial.
