  The feasibility of observing ditauonium, the bound state of two tau leptons,
at $e^+e^-$ colliders (BES III at $\sqrt{s} = 3.78$ GeV, Belle II at $\sqrt{s}
= 10.6$ GeV, a future super tau-charm factory (STCF) at $\sqrt{s} \approx
2m_{\tau}$, and the FCC-ee at $\sqrt{s} = 91.2$ GeV) as well as in hadronic and
photon-photon collisions at the LHC, is studied. Cross sections and expected
yields for spin-0 para- ($\mathcal{T}_0$) and spin-1 ortho- ($\mathcal{T}_1$)
ditauonium are presented for nine different production and decay processes.
Para-ditauonium can be observed at the FCC-ee via photon fusion in its diphoton
decay ($\gamma\gamma\to\mathcal{T}_0\to\gamma\gamma$). Ortho-ditauonium can be
observed at STCF via $e^+e^-\to\mathcal{T}_1\to\mu^+\mu^-$, where a threshold
scan with monochromatized beams can also provide a very precise extraction of
the tau lepton mass with a $\mathcal{O}(25$ keV) uncertainty or less. Observing
pp $\to \mathcal{T}_1(\mu^+\mu^-)+X$ at the LHC is possible by identifying its
displaced vertex with a good control of the combinatorial dimuon background. In
addition, we compute the rare decay branching fractions of ditauonium into
quarkonium plus a photon.
