  A search for the associated production of the Higgs boson with a top quark
pair ($t\bar t H$) is reported. The search is performed in multilepton final
states using a dataset corresponding to an integrated luminosity of 36.1
fb$^{-1}$ of proton--proton collision data recorded by the ATLAS experiment at
a center-of-mass energy $\sqrt{s} = 13$ TeV at the Large Hadron Collider. Higgs
boson decays to $WW^*$, $\tau\tau$, and $ZZ^*$ are targeted. Seven final
states, categorized by the number and flavor of charged-lepton candidates, are
examined for the presence of the Standard Model Higgs boson with a mass of 125
GeV and a pair of top quarks. An excess of events over the expected background
from Standard Model processes is found with an observed significance of 4.1
standard deviations, compared to an expectation of 2.8 standard deviations. The
best fit for the $t\bar t H$ production cross section is $\sigma(t\bar t H) =
790^{+230}_{-210}$ fb, in agreement with the Standard Model prediction of
$507^{+35}_{-50}$ fb. The combination of this result with other $t\bar t H$
searches from the ATLAS experiment using the Higgs boson decay modes to $b\bar
b$, $\gamma\gamma$ and $ZZ^* \to 4\ell$, has an observed significance of 4.2
standard deviations, compared to an expectation of 3.8 standard deviations.
This provides evidence for the $t\bar t H$ production mode.
