  We present the Lyman-$\alpha$ (Ly$\alpha$) and ultraviolet (UV) luminosity
function (LF), in bins of redshift, of 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 $\sim 36$ deg$^2$ observed in 40
narrow-band filters (NB; FWHM $\sim 120$ \r{A}). We cover the
intermediate-bright luminosity regime of the LF $(10^{43.5}<(L_{{\rm
Ly}\alpha}/{\rm erg\,s}^{-1})<10^{45.5}$; $-29<M_{\rm UV}<-24)$. The continuous
wavelength coverage of the PAUS NB set allows a very efficient target
identification and precise redshift measurements. We show that our method is
able to retrieve a fairly complete ($C\sim 85\%$) and pure ($P\sim 90\%$)
sample of Ly$\alpha$ emitting quasars for $L_{{\rm Ly}\alpha}>10^{44}$ ${\rm
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 Ly$\alpha$ and UV LFs,
with a declining tendency in the number density of quasars towards increasing
redshifts. In addition, the faint-end power-law slope of the Ly$\alpha$ LF
becomes steeper with redshift, suggesting that the number density of
Ly$\alpha$-bright quasars declines faster than that of fainter emitters. By
integrating the Ly$\alpha$ LF we find that the total Ly$\alpha$ emitted by
bright quasars per unit volume rapidly declines with increasing redshift, being
sub-dominant to that of star-forming galaxies by several orders of magnitude by
$z\sim 4$. 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 to measure the mean IGM absorption by the Lyman-$\alpha$
forest as a function of redshift.
