  The Ces\`aro function spaces $Ces_p=[C,L^p]$, $1\le p\le\infty$, have
received renewed attention in recent years. Many properties of $[C,L^p]$ are
known. Less is known about $[C,X]$ when the Ces\`aro operator takes its values
in a rearrangement invariant (r.i.) space $X$ other than $L^p$. In this paper
we study the spaces $[C,X]$ via the methods of vector measures and vector
integration. These techniques allow us to identify the absolutely continuous
part of $[C,X]$ and the Fatou completion of $[C,X]$; to show that $[C,X]$ is
never reflexive and never r.i.; to identify when $[C,X]$ is weakly sequentially
complete, when it is isomorphic to an AL-space, and when it has the
Dunford-Pettis property. The same techniques are used to analyze the operator
$C:[C,X]\to X$; it is never compact but, it can be completely continuous.
