We present the Lyman-α (Lyα) and ultraviolet (UV) luminosity function (LF) in bins of redshift for quasars selected in the Physics of the Accelerating Universe Survey (PAUS). A sample of 915 objects was selected at 2.7 < z < 5.3 within an effective area of ∼36 deg2 observed in 40 narrow-band (NB) filters (FWHM ∼ 120 Å). We cover the intermediate–bright luminosity regime of the LF ($ 10^{43.5} < (L_{\mathrm{Ly}\alpha}/\mathrm{erg\,s}^{-1}) < 10^{45.5} $; −29 < MUV < −24). The continuous wavelength coverage of the PAUS NB set allows very efficient target identification and precise redshift measurements. We show that our method is able to retrieve a relatively complete (C ∼ 85%) and pure (P ∼ 90%) sample of Lyα-emitting quasars for LLyα > 1044 erg s−1. In order to obtain corrections for the LF estimation, and assess the accuracy of our selection method, we produced mock catalogs of 0 < z < 4.3 quasars and galaxies that mimic our target population and their main contaminants. Our results show a clear evolution of the Lyα and UV LFs, with a declining tendency in the number density of quasars toward increasing redshifts. In addition, the faint-end power-law slope of the Lyα LF becomes steeper with redshift, suggesting that the number density of Lyα-bright quasars declines faster than that of fainter emitters. By integrating the Lyα LF, we find that the total Lyα emitted by bright quasars per unit volume rapidly declines with increasing redshift, being subdominant to that of star-forming galaxies by several orders of magnitude by z ∼ 4. Finally, we stack the NB pseudo-spectra of a visually selected “golden sample” of 591 quasars to obtain photometric composite SEDs in bins of redshift, enabling us to measure the mean intergalactic medium absorption using the Lyman-α forest as a function of redshift, yielding results consistent with previous spectroscopic determinations.