The measurement of the energy spectra and densities of α-particles and other fast ions are part of the ITER measurement requirements, highlighting the importance of energy-resolved energetic-particle measurements for the mission of ITER. However, it has been found in recent years that the velocity-space interrogation regions of the foreseen energetic-particle diagnostics do not allow these measurements directly. We will demonstrate this for γ-ray spectroscopy (GRS), collective Thomson scattering (CTS), neutron emission spectroscopy and fast-ion Dα spectroscopy by invoking energy and momentum conservation in each case, highlighting analogies and differences between the different diagnostic velocity-space sensitivities. Nevertheless, energy spectra and densities can be inferred by velocity-space tomography which we demonstrate using measurements at JET and ASDEX Upgrade. The measured energy spectra agree well with corresponding simulations. At ITER, α-particle energy spectra and densities can be inferred for energies larger than 1.7 MeV by velocity-space tomography based on GRS and CTS. Further, assuming isotropy of the α-particles in velocity space, their energy spectra and densities can be inferred by 1D inversion of spectral single-detector measurements down to about 300 keV by CTS. The α-particle density can also be found by fitting a model to the CTS measurements assuming the α-particle distribution to be an isotropic slowing-down distribution.