It is known since 1954 that every $3$-manifold bounds a $4$-manifold. Thus, for instance, every $3$-manifold has a surgery diagram. There are many proofs of this fact, including several constructive ones, but they do not bound the complexity of the $4$-manifold. Given a $3$-manifold $M$ of complexity $n$, we show how to construct a $4$-manifold bounded by $M$ of complexity $O(n^2)$, for suitable notions of ``complexity". It is an open question whether this quadratic bound can be replaced by a linear bound. \vskip .1in \noindent The natural setting for this result is shadow surfaces, a representation of $3$- and $4$-manifolds that generalizes many other representations of these manifolds. One consequence of our results is some intriguing connections between the complexity of a shadow representation and the hyperbolic volume of a $3$-manifold. \vskip .1in \noindent (Joint work with Francesco Costantino.)