This paper describes a computer study of tokamak equilibria, which resulted in configurations at β-values close to one. The cross-section that was studied most intensely was a vertical racetrack with an elongation of two to one. A circular cross-section and a D-shaped vessel did not allow equilibria at as high a β as the racetrack. The safety factor q was kept greater than one everywhere, and the q-profile was kept flat. The corresponding broad current profile has been observed in pinches, where it has a very favourable influence on equilibrium and stability. This favourable influence also displays itself in the equilibria computed; even at a peak β of 82%, the magnetic axis shifts over only 37% of the half-width of the cross-section. The outer boundary of the cross-section was assumed to be a flux surface (e.g. a metal shell). Stability of localized modes was studied by means of the Mercier criterion; it was possible to satisfy this criterion, except in the vicinity of the magnetic axis. Calculations with narrow pressure profiles showed that the plasma did not contract to a round shape; the broad current profile apparently helps to maintain the elongated shape of the plasma. The very high β-values reported here are of interest for extrapolations of fast pulsed tokamaks. It is also important to note that even a modest broadening of the current profile in a conventional tokamak would improve the stability and reduce the shift at moderate β.