  A $k$-block in a graph $G$ is a maximal set of at least $k$ vertices no two
of which can be separated in $G$ by deleting fewer than $k$ vertices. The block
number $\beta(G)$ of $G$ is the maximum integer $k$ for which $G$ contains a
$k$-block.
  We prove a structure theorem for graphs without a $(k+1)$-block, showing that
every such graph has a tree-decomposition in which every torso has at most $k$
vertices of degree $2k^2$ or greater. This yields a qualitative duality, since
every graph that admits such a decomposition has block number at most $2k^2$.
  We also study $k$-blocks in graphs from classes of graphs $\mathcal{G}$ that
exclude some fixed graph as a topological minor, and prove that every $G \in
\mathcal{G}$ satisfies $\beta(G) \leq c\sqrt[3]{|G|}$ for some constant $c = c(
\mathcal{G})$.
  Moreover, we show that every graph of tree-width at least $2k^2$ has a minor
containing a $k$-block. This bound is best possible up to a multiplicative
constant.
