  Given a branching random walk, let $M_n$ be the minimum position of any
member of the $n$th generation. We calculate $\mathbf{E}M_n$ to within O(1) and
prove exponential tail bounds for $\mathbf{P}\{|M_n-\mathbf{E}M_n|>x\}$, under
quite general conditions on the branching random walk. In particular, together
with work by Bramson [Z. Wahrsch. Verw. Gebiete 45 (1978) 89--108], our results
fully characterize the possible behavior of $\mathbf {E}M_n$ when the branching
random walk has bounded branching and step size.
