Asymptotic dimension of metric spaces is a large-scale analog of covering dimension of topological spaces. An intersection graph of a family of sets is the graph whose vertices are the members of the family and whose edges correspond to pairs of members with non-empty intersection. Our first main result connects the asymptotic dimension of the intersection graph of a family ${\mathcal F}$ and the Assouad-Nagata dimension of the ambient metric space containing members of ${\mathcal F}$ under some mild and necessary assumptions. We prove that if ${\mathcal F}$ is a family of subsets of a metric space of Assouad-Nagata dimension $n$ such that every ball of radius $r$ intersects at most $f(r/s)$ pairwise disjoint members of ${\mathcal F}$ of diameter at least $s$ for some function $f$, then the asymptotic dimension of the intersection graph of ${\mathcal F}$ is at most $n+1$. This result is optimal both quantitatively and qualitatively in several senses. As a corollary of this result, the asymptotic dimension of the intersection graph of any family of compact convex sets of bounded aspect ratio in ${\mathbb R}^n$, such as a family of balls in ${\mathbb R}^n$, is at most $n+1$. Our second main result states that the asymptotic dimension of the intersection graph of a family ${\mathcal F}$ of connected closed sets of a connected topological space with connected boundary equals the asymptotic dimension of the intersection graph of the family of the boundary of the sets in ${\mathcal F}$, under a mild condition. In particular, the asymptotic dimension of the intersection graphs of families of spheres in ${\mathbb R}^n$ equals $n$ or $n+1$ when $n \geq 2$.

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