  Let $G$ be a finite simple group. By a theorem of Guralnick and Kantor, $G$
contains a conjugacy class $C$ such that for each non-identity element $x \in
G$, there exists $y \in C$ with $G = \langle x,y\rangle$. Building on this deep
result, we introduce a new invariant $\gamma_u(G)$, which we call the uniform
domination number of $G$. This is the minimal size of a subset $S$ of conjugate
elements such that for each $1 \ne x \in G$, there exists $s \in S$ with $G =
\langle x, s \rangle$. (This invariant is closely related to the total
domination number of the generating graph of $G$, which explains our choice of
terminology.) By the result of Guralnick and Kantor, we have $\gamma_u(G)
\leqslant |C|$ for some conjugacy class $C$ of $G$, and the aim of this paper
is to determine close to best possible bounds on $\gamma_u(G)$ for each family
of simple groups. For example, we will prove that there are infinitely many
non-abelian simple groups $G$ with $\gamma_u(G) = 2$. To do this, we develop a
probabilistic approach, based on fixed point ratio estimates. We also establish
a connection to the theory of bases for permutation groups, which allows us to
apply recent results on base sizes for primitive actions of simple groups.
