  We exploit the excursion set approach in integral formulation to derive
novel, accurate analytic approximations of the unconditional and conditional
first crossing distributions, for random walks with uncorrelated steps and
general shapes of the moving barrier; we find the corresponding approximations
of the unconditional and conditional halo mass functions for Cold Dark Matter
power spectra to represent very well the outcomes of state-of-the-art
cosmological N-body simulations. In addition, we apply these results to derive,
and confront with simulations, other quantities of interest in halo statistics,
including the rates of halo formation and creation, the average halo growth
history, and the halo bias. Finally, we discuss how our approach and main
results change when considering random walks with correlated instead of
uncorrelated steps, and Warm instead of Cold Dark Matter power spectra.

Cosmology and Nongalactic Astrophysics