  (Abridged) The ESO Slice Project (ESP) is a galaxy redshift survey we have
completed as an ESO Key-Project over ~23 square degrees, in a region near the
South Galactic Pole. The survey is nearly complete to the limiting magnitude
b_J=19.4 and consists of 3342 galaxies with reliable redshift determination.
The ESP survey is intermediate between shallow, wide angle samples and very
deep, one-dimensional pencil beams: spanning a volume of ~ 5 x 10^4 Mpc^3 at
the sensitivity peak (z ~ 0.1), it provides an accurate determination of the
"local" luminosity function and the mean galaxy density. We find that, although
a Schechter function is an acceptable representation of the luminosity function
over the entire range of magnitudes (M < -12.4), our data suggest the presence
of a steepening of the luminosity function for M > -17. The amplitude and the
alpha and M^* parameters of our luminosity function are in good agreement with
those of the AUTOFIB redshift survey (Ellis et al. 1996). Viceversa, our
amplitude is significantly higher, by a factor ~ 1.6 at M ~ M^*, than that
found for both the Stromlo-APM (Loveday et al. 1992) and the Las Campanas (Lin
et al. 1996) redshift surveys. Also the faint end slope of our luminosity
function is significantly steeper than that found in these two surveys.
  Large over- and under- densities are clearly seen in our data. In particular,
we find evidence for a "local" underdensity (for D < 140 Mpc) and a significant
overdensity at z ~ 0.1. When these radial density variations are taken into
account, our derived luminosity function reproduces very well the observed
counts for b_J < 19.4, including the steeper than Euclidean slope for b_J < 17.
