  We report on VLBI observations of the fast and blue optical transient (FBOT),
AT 2018cow. At ~62 Mpc, AT 2018cow is the first relatively nearby FBOT. The
nature of AT 2018cow is not clear, although various hypotheses from a tidal
disruption event to different kinds of supernovae have been suggested. It had a
very fast rise time (3.5 d) and an almost featureless blue spectrum although
high photospheric velocities (40,000 km s$^{-1}$) were suggested early on. The
X-ray luminosity was very high, ~$1.4 \times 10^{43}$ erg s$^{-1}$, larger than
those of ordinary SNe, and more consistent with those of SNe associated with
gamma-ray bursts. Variable hard X-ray emission hints at a long-lived "central
engine." It was also fairly radio luminous, with a peak 8.4-GHz spectral
luminosity of ~$4 \times 10^{28}$ erg s$^{-1}$ Hz$^{-1}$, allowing us to make
VLBI observations at ages between 22 and 287 d. We do not resolve AT 2018cow.
Assuming a circularly symmetric source, our observations constrain the average
apparent expansion velocity to be <0.49$c$ by t = 98 d (3$\sigma$ limit). We
also constrain the proper motion of AT 2018cow to be <0.51$c$. Since the radio
emission generally traces the fastest ejecta, our observations make the
presence of a long-lived relativistic jet with a lifetime of more than one
month very unlikely.
