  The maximum number of vertices in a graph of maximum degree $\Delta\ge 3$ and
fixed diameter $k\ge 2$ is upper bounded by $(1+o(1))(\Delta-1)^{k}$. If we
restrict our graphs to certain classes, better upper bounds are known. For
instance, for the class of trees there is an upper bound of
$(2+o(1))(\Delta-1)^{\lfloor k/2\rfloor}$ for a fixed $k$. The main result of
this paper is that graphs embedded in surfaces of bounded Euler genus $g$
behave like trees, in the sense that, for large $\Delta$, such graphs have
orders bounded from above by \[begin{cases} c(g+1)(\Delta-1)^{\lfloor
k/2\rfloor} & \text{if $k$ is even} c(g^{3/2}+1)(\Delta-1)^{\lfloor k/2\rfloor}
& \text{if $k$ is odd}, \{cases}\] where $c$ is an absolute constant. This
result represents a qualitative improvement over all previous results, even for
planar graphs of odd diameter $k$. With respect to lower bounds, we construct
graphs of Euler genus $g$, odd diameter $k$, and order
$c(\sqrt{g}+1)(\Delta-1)^{\lfloor k/2\rfloor}$ for some absolute constant
$c>0$. Our results answer in the negative a question of Miller and
\v{S}ir\'a\v{n} (2005).
