  We develop a class of exponential bounds for the probability that a
martingale sequence crosses a time-dependent linear threshold. Our key insight
is that it is both natural and fruitful to formulate exponential concentration
inequalities in this way. We illustrate this point by presenting a single
assumption and theorem that together unify and strengthen many tail bounds for
martingales, including classical inequalities (1960-80) by Bernstein, Bennett,
Hoeffding, and Freedman; contemporary inequalities (1980-2000) by Shorack and
Wellner, Pinelis, Blackwell, van de Geer, and de la Pe\~na; and several modern
inequalities (post-2000) by Khan, Tropp, Bercu and Touati, Delyon, and others.
In each of these cases, we give the strongest and most general statements to
date, quantifying the time-uniform concentration of scalar, matrix, and
Banach-space-valued martingales, under a variety of nonparametric assumptions
in discrete and continuous time. In doing so, we bridge the gap between
existing line-crossing inequalities, the sequential probability ratio test, the
Cram\'er-Chernoff method, self-normalized processes, and other parts of the
literature.
