  The zero sets of the Bergman space $A^p_\omega$ induced by either a radial
weight $\omega$ admitting a certain doubling property or a non-radial
Bekoll\'e-Bonami type weight are characterized in the spirit of Luecking's
results from 1996. Accurate results obtained en route to this characterization
are used to generalize Horowitz's factorization result from 1977 for functions
in $A^p_\omega$. The utility of the obtained factorization is illustrated by
applications to integration and composition operators as well as to small
Hankel operator induced by a conjugate analytic symbol. Dominating sets and
sampling measures for the weighted Bergman space $A^p_\omega$ induced by a
doubling weight are also studied. Several open problems related to the scheme
of the paper are posed.
