  We study the production of a top-quark pair in association with one and two
vector bosons, $t \bar t V$ and $t \bar t VV$ with $V=\gamma, Z, W^\pm$, at the
LHC. We provide predictions at next-to-leading order in QCD for total cross
sections and top-quark charge asymmetries as well as for differential
distributions. A thorough discussion of the residual theoretical uncertainties
related to missing higher orders and to parton distribution functions is
presented. As an application, we calculate the total cross sections for this
class of processes (together with $t \bar t H$ and $t \bar t t \bar t$
production) at hadron colliders for energies up to 100 TeV. In addition, by
matching the NLO calculation to a parton shower, we determine the contribution
of $t \bar t V$ and $t \bar t VV$ to final state signatures (two-photon and
two-same-sign-, three- and four-lepton) relevant for $t \bar t H$ analyses at
the Run II of the LHC.
