We study the maximal displacement of a one dimensional subcritical branching random walk initiated by a single particle at the origin. For each ∈ℕ, let be the rightmost position reached by the branching random walk up to generation n. Under the assumption that the offspring distribution has a finite third moment and the jump distribution has mean zero and a finite probability generating function, we show that there exists >1 such that the function
( , ):= ( ≥ ),for each >0 and ∈ℕ,
satisfies the following properties: there exist 0< ⎯⎯≤ ⎯⎯⎯<∞ such that if < ⎯⎯, then
0<liminf →∞ ( , )≤limsup →∞ ( , )≤1,
while if > ⎯⎯⎯, then
lim →∞ ( , )=0.
Moreover, if the jump distribution has a finite right range R, then ⎯⎯⎯< . If furthermore the jump distribution is “nearly right-continuous”, then there exists ∈(0,1] such that lim →∞ ( , )= for all < ⎯⎯. We also show that the tail distribution of :=sup ≥0 , namely, the rightmost position ever reached by the branching random walk, has a similar exponential decay (without the cutoff at ⎯⎯). Finally, by duality, these results imply that the maximal displacement of supercritical branching random walks conditional on extinction has a similar tail behavior.