  Context: Vela X-1 is one of the best studied X-ray binaries. Frequently
though, specific values for its parameters have been used in subsequent studies
without considering alternatives. Aims: We aim to provide a robust compilation
and synthesis of the accumulated knowledge about Vela X-1 as a solid baseline
for future studies and identify specific avenues of possible future research.
Methods: We explore the literature for Vela X-1 and on modelling efforts,
describing the evolution of the system knowledge. We also add information
derived from public data, especially the Gaia EDR3 release. Results: We update
the distance to Vela X-1, the spectral classification for HD 77518 and find
that the supergiant may be very close to filling its Roche lobe. Constraints on
the clumpiness of the stellar wind have improved. The orbit is very well
determined, but the uncertain inclination limits information on the neutron
star mass. Estimates for the stellar wind have evolved towards lower
velocities, supporting the idea of transient wind-captured disks around the
neutron star. Hydrodynamic models and observations are consistent with an
accretion wake trailing the neutron star. Conclusions: Vela X-1 is an excellent
laboratory, but a lot of room remains to improve. Well-coordinated
multi-wavelength observations and campaigns addressing the intrinsic
variability are required. New opportunities will arise through new
instrumentation. Models of the stellar wind should account for the orbital
eccentricity and the non-spherical shape of HD 77581. Realistic
multi-dimensional models of radiative transfer in the UV and X-rays are needed,
but remain very challenging. Improved MHD models covering a wide range of
scales would be required to improve understanding of the plasma-magnetosphere
coupling. A full characterization of the accretion column remains another open
challenge. (Abbreviated for arXiv)
