  In the past decades, most work in the area of data analysis and machine
learning was focused on optimizing predictive models and getting better results
than what was possible with existing models. To what extent the metrics with
which such improvements were measured were accurately capturing the intended
goal, whether the numerical differences in the resulting values were
significant, or whether uncertainty played a role in this study and if it
should have been taken into account, was of secondary importance. Whereas
probability theory, be it frequentist or Bayesian, used to be the gold standard
in science before the advent of the supercomputer, it was quickly replaced in
favor of black box models and sheer computing power because of their ability to
handle large data sets. This evolution sadly happened at the expense of
interpretability and trustworthiness. However, while people are still trying to
improve the predictive power of their models, the community is starting to
realize that for many applications it is not so much the exact prediction that
is of importance, but rather the variability or uncertainty.
  The work in this dissertation tries to further the quest for a world where
everyone is aware of uncertainty, of how important it is and how to embrace it
instead of fearing it. A specific, though general, framework that allows anyone
to obtain accurate uncertainty estimates is singled out and analysed. Certain
aspects and applications of the framework -- dubbed `conformal prediction' --
are studied in detail. Whereas many approaches to uncertainty quantification
make strong assumptions about the data, conformal prediction is, at the time of
writing, the only framework that deserves the title `distribution-free'. No
parametric assumptions have to be made and the nonparametric results also hold
without having to resort to the law of large numbers in the asymptotic regime.
