We present calculations of higher-order QCD and electroweak (EW) corrections to the associated production of a top-antitop quark pair and a $Z$ boson, i.e. $t{\bar t}Z$ production, in proton collisions. We find that the contributions from soft-gluon corrections are numerically dominant and large. We present approximate NNLO (aNNLO) and approximate N$^3$LO (aN$^3$LO) cross sections that include soft corrections on top of the exact NLO QCD result. We also add electroweak corrections through NLO. We compare our aN$^3$LO QCD + NLO EW theoretical results to measurements of total cross sections from the LHC, and we also calculate top-quark differential distributions in transverse momentum and rapidity.