  This paper presents the observation of four-top-quark ($t\bar{t}t\bar{t}$)
production in proton-proton collisions at the LHC. The analysis is performed
using an integrated luminosity of 140 fb$^{-1}$ at a centre-of-mass energy of
13 TeV collected using the ATLAS detector. Events containing two leptons with
the same electric charge or at least three leptons (electrons or muons) are
selected. Event kinematics are used to separate signal from background through
a multivariate discriminant, and dedicated control regions are used to
constrain the dominant backgrounds. The observed (expected) significance of the
measured $t\bar{t}t\bar{t}$ signal with respect to the standard model (SM)
background-only hypothesis is 6.1 (4.3) standard deviations. The
$t\bar{t}t\bar{t}$ production cross section is measured to be
$22.5^{+6.6}_{-5.5}$ fb, consistent with the SM prediction of $12.0 \pm 2.4$ fb
within 1.8 standard deviations. Data are also used to set limits on the
three-top-quark production cross section, being an irreducible background not
measured previously, and to constrain the top-Higgs Yukawa coupling and
effective field theory operator coefficients that affect $t\bar{t}t\bar{t}$
production.
