  The question of when the Bergman projection $P_\omega$ induced by a radial
weight $\omega$ on the unit disc is a bounded operator from one space into
another is of primordial importance in the theory of Bergman spaces. The
long-standing problem of describing the radial weights $\omega$ such that
$P_\omega$ is bounded on the Lebesgue space $L^p_\omega$ had been known to
experts since decades before it was formally posed by Dostani\'c in 2004. A
natural limit case of this setting is when $P_\omega$ acts from $L^\infty$ to
the Bloch space. The surjectivity of the operator becomes another relevant
question in this limit case.
  The main findings of this study are shortly listed as follows. We establish
characterizations of the radial weights $\omega$ on the unit disc such that
$P_\omega:L^\infty\to\mathcal{B}$ is bounded and/or acts surjectively, or the
dual of $A^1_\omega$ is isomorphic to the Bloch space $\mathcal{B}$ under the
$A^2_\omega$-pairing. We also solve the problem posed by Dostani\'c under a
weak regularity hypothesis on the weight involved. With regard to
Littlewood-Paley estimates, we describe the radial weights $\omega$ such that
the norm of any function in $A^p_\omega$ is comparable to the norm in
$L^p_\omega$ of its derivative times the distance from the boundary. This
last-mentioned result solves another well-known problem on the area. All
characterizations can be given in terms of doubling conditions on moments
and/or tail integrals $\int_r^1\omega(t)\,dt$ of $\omega$, and are therefore
easy to interpret.
  We also make substantial progress about the two weight inequality
  $$
  \|P_\omega(f)\|_{L^p_\nu}\le C\|f\|_{L^p_\nu},\quad f\in L^p_\nu, \quad
1<p<\infty.
  $$ for radial weights $\omega$ and $\nu$.
