We present a study of the evolution of active galactic nuclei (AGN) selected in Einstein and ROSAT surveys. A total of 748 broad-line AGN are included in the study, including 29 with z>2. Pure luminosity evolution provides a good description of the evolution for q_0=0. For q_0=0.5 we find that pure luminosity evolution underpredicts the number of intermediate-redshift (1.4<z<2), low-luminosity AGN. Evolution models with strong evolution at high redshift (z>1.8) are rejected at high confidence levels. Models with no evolution or negative evolution at z>1.8 are acceptable. <V_e/V_a> tests, which are model-independent, suggest that at z>1.8 evolution is negative, i.e., QSO space densities and/or luminosities decline with redshift. Our model luminosity functions predict that broad-line AGN with z<4 contribute less than 45 per cent of the 1-2 keV X-ray background. We derive a simple expression to correct a two-power-law luminosity function for the spurious evolution caused by the dispersion of X-ray spectral slopes. When the dispersion in AGN spectral slopes is taken into account, we find that the redshift at which evolution stops may be as low as z~1.4 for both q_0=0 and q_0=0.5.