  An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and
$\prec$ is a total ordering of its vertices. The ordered Ramsey number
$\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every ordered
complete graph with $N$ vertices and with edges colored by two colors contains
a monochromatic copy of $\mathcal{G}$.
  In contrast with the case of unordered graphs, we show that there are
arbitrarily large ordered matchings $\mathcal{M}_n$ on $n$ vertices for which
$\overline{R}(\mathcal{M}_n)$ is superpolynomial in $n$. This implies that
ordered Ramsey numbers of the same graph can grow superpolynomially in the size
of the graph in one ordering and remain linear in another ordering.
  We also prove that the ordered Ramsey number $\overline{R}(\mathcal{G})$ is
polynomial in the number of vertices of $\mathcal{G}$ if the bandwidth of
$\mathcal{G}$ is constant or if $\mathcal{G}$ is an ordered graph of constant
degeneracy and constant interval chromatic number. The first result gives a
positive answer to a question of Conlon, Fox, Lee, and Sudakov.
  For a few special classes of ordered paths, stars or matchings, we give
asymptotically tight bounds on their ordered Ramsey numbers. For so-called
monotone cycles we compute their ordered Ramsey numbers exactly. This result
implies exact formulas for geometric Ramsey numbers of cycles introduced by
K\'arolyi, Pach, T\'oth, and Valtr.
