We establish characterizations of the radial weights $\omega$ on the unit disc such that the Bergman projection $P_\omega$, induced by $\omega$, is bounded and/or acts surjectively from $L^\infty$ to the Bloch space $\mathcal{B}$, or the dual of the weighted Bergman space $A^1_\omega$ is isomorphic to the Bloch space under the $A^2_\omega$-pairing. We also solve the problem posed by Dostani\'c in 2004 of describing the radial weights~$\omega$ such that~$P_\omega$ is bounded on the Lebesgue space~$L^p_\omega$, under a weak regularity hypothesis on the weight involved. With regard to Littlewood-Paley estimates, we characterize 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.
