Selective inference methods have garnered considerable attention in the past few years. One of them, the so-called conditional approach, under which inferential statements are derived from a conditional model of the data, provides a particularly simple and widely applicable solution to the problem. The aim of this thesis is to carry out a detailed analysis of this approach from several angles. We analyse it conceptually from frequentist and Bayesian viewpoints, establish some theoretical results, and propose routes for numerical implementation of the approach in complex regimes. Two key components of this work are the derivation of a class of approximate probability-matching priors for selection models, and the analysis of a simple alternative to data splitting in regression models which produces more powerful inferences than the latter method. We provide an empirical validation of all the methodological proposals.