  In this thesis we analyse three aspects of Conformal Field Theories (CFTs).
First, we consider correlation functions of descendant states in
two-dimensional CFTs. We discuss a recursive formula to calculate them and
provide a computer implementation of it. This allows us to obtain any
correlation function of vacuum descendants, and for non-vacuum descendants to
express the correlator as a differential operator acting on the respective
primary correlator. With this code, we study some entanglement and
distinguishability measures between descendant states, i.e. the R\'enyi
entropy, trace square distance and sandwiched R\'enyi divergence. With our
results we can test the R\'enyi Quantum Null Energy Condition and provide new
tools to analyse the holographic description of descendant states. Second, we
study four-dimensional Weyl fermions on different backgrounds. Our interest is
in their trace anomaly, where the Pontryagin density has been claimed to
appear. To ascertain this possibility, we compute the anomalies of Dirac
fermions coupled to vector and axial non-abelian gauge fields and then in a
metric-axial-tensor background. Appropriate limits of the backgrounds allow to
recover the anomalies of Weyl fermions coupled to non-abelian gauge fields and
in a curved spacetime. In both cases we confirm the absence of the Pontryagin
density in the trace anomalies. Third, we provide the holographic description
of a four-dimensional CFT with an irrelevant operator. When the operator has
integer conformal dimension, its presence in the CFT modifies the Weyl
transformation of the metric, which in turns modifies the trace anomaly.
Exploiting the equivalence between bulk diffeomorphisms and boundary Weyl
transformations, we compute these modifications from the dual gravity theory.
Our results represent an additional test of the AdS/CFT conjecture.
