We present a model for the cosmological evolution of quasars (QSOs) under the assumption that they are powered by massive accreting black holes. Accretion flows around massive black holes make a transition from high radiative efficiency ($\sim 10%$) to low efficiency, advection-dominated flows when ${\dot M}/{\dot M}_{Edd}$ falls below the critical rate $\sim 0.3α^2\sim 10^{-2}$ where ${\dot M}$ is the mass accretion rate, ${\dot M}_{Edd}\propto M$ is the usual Eddington rate with the nominal 10% efficiency, and $α(\le 1)$ is the dimensionless viscosity parameter. We identify this transition with the observed break at a redshift $\sim 2$ in the QSOs' X-ray luminosity evolution. Growth of black holes through accretion could naturally lead to such a transition at a critical redshift $z_c\sim 1-3$, provided that most of high redshift QSOs appear with near Eddington luminosities at $z\sim 3-4$ and the accretion rates decline over the Hubble time in a roughly synchronous manner. Before the transition, the QSOs' luminosities (with a high efficiency) slowly decrease and after the transition at $z_c$, the QSO luminosities evolve approximately as $\propto (1+z)^{K(z)}$ where $K(z)$ gradually varies from $z=z_c$ to $z\sim 0$ around $K\sim3$. The results depend on the details of the QSO X-ray emission mechanism. We discuss some further implications.

Latex, 2 PS figures, To Appear in ApJ