  We investigate extreme value statistics (EVS) of general discrete time and
continuous space symmetric jump processes. We first show that for unbounded
jump processes, the semi-infinite propagator $G_0(x,n)$, defined as the
probability for a particle issued from $0$ to be at position $x$ after $n$
steps whilst staying positive, is the key ingredient needed to derive a variety
of joint distributions of extremes and times at which they are reached. Along
with exact expressions, we extract novel universal asymptotic behaviors of such
quantities. For bounded, semi-infinite jump processes killed upon first
crossing of zero, we introduce the \textit{strip probability}
$\mu_{0,\underline{x}}(n)$, defined as the probability that a particle issued
from 0 remains positive and reaches its maximum $x$ on its $n^{\rm th}$ step
exactly. We show that $\mu_{0,\underline{x}}(n)$ is the essential building
block to address EVS of semi-infinite jump processes, and obtain exact
expressions and universal asymptotic behaviors of various joint distributions.
