Abstract <p>Mass spectra of triply and fully heavy tetraquarks were calculated within the relativistic quark model, based on the quasipotential approach and QCD. A tetraquark is considered in the framework of the diquark–antidiquark picture as a bound state of a diquark and an antidiquark which are assumed to be in a ground state. Diquarks are treated as color triplet states. Calculations take into account relativistic effects and finite size of diquarks. Obtained results are compared with the strong fall–apart decay thresholds into a pair of mesons. Recent data on observations of resonances in charmonium pair production are discussed. Tetraquark states that are most promising candidates for experimental searches are identified.</p>