  We classify smooth weak del Pezzo surfaces with global vector fields over an
arbitrary algebraically closed field $k$ of arbitrary characteristic $p \geq
0$. We give a complete description of the configuration of $(-1)$- and
$(-2)$-curves on these surfaces and calculate the identity component of their
automorphism schemes. It turns out that there are $53$ distinct families of
such surfaces if $p \neq 2,3$, while there are $61$ such families if $p = 3$,
and $75$ such families if $p = 2$. Each of these families has at most one
moduli. As a byproduct of our classification, it follows that weak del Pezzo
surfaces with non-reduced automorphism scheme exist over $k$ if and only if $p
\in \{2,3\}$.
