  The implications of seven popular models of quintessence based on
supergravity or M/string theory for the transition from a decelerating to an
accelerating universe are explored. All seven potentials can mimic the
LambdaCDM model at low redshifts 0 < z < 5. However, for a natural range of
initial values of the quintessence field, the SUGRA and Polonyi potentials
predict a transition redshift z_t = 0.5 for Omega_Lambda0 = 0.70, in agreement
with the observational value z_t = 0.46 and in mild conflict with the LambdaCDM
value z_t = 0.67. Tables are given for the quintessence potentials for the
recent average wbar_0 of the equation of state parameter, and for w_0 and w_1
in the low-z approximation w = w_0 + w_1 z. It is argued that for the
exponential potential e^{lambda phi} to produce a viable present-day cosmology,
lambda <= sqrt{2}. A robust, scaled numerical method is presented for
simulating the cosmological evolution of the scalar field.
