This paper is devoted to the general study of the Borel and analytic sets, in products of two Polish Spaces, the sections of which possess a given property $\mathbb{P}$. We are particularly interested in separation results for analytic sets, and in the following "Borel selection" type of problems : Let $\mathbb{P}_{\sigma}$ be the property of being the countable union of sets having property $\mathbb{P}$. What conditions on $\mathbb{P}$ imply that every Borel set with sections possessing property $\mathbb{P}_{\sigma}$ is the union of a sequence of Borel sets with sections possessing property $\mathbb{P}$ ?
The paper contains numerous applications of the general theory, as well as a discussion of possible extensions to abstract measured spaces, and to higher levels of the projective hierarchy.
The methods used in the proofs being those of Effective Descriptive Set Theory, we also present the basic notions of this theory necessary for our purposes.