  In this thesis, we analyze unitary conformal field theories in three
dimensional spaces by applying analytic conformal bootstrap techniques to
correlation functions of non-scalar operators, in particular Majorana fermions.
Via the analysis of these correlation functions, we access several sectors in
the spectrum of conformal field theories that have been previously unexplored
with analytic methods, and we provide new data for several operator families.
In the first part of the thesis, we achieve this by the large spin expansions
that have been traditionally used in the conformal bootstrap program for scalar
correlators, whereas in the second part we carry out the computations by
combining several analytic tools that have been recently developed such as
weight shifting operators, harmonic analysis for the Euclidean conformal group,
and the Lorentzian inversion formula. We compare these methods and demonstrate
the superiority of the latter by computing nonperturbative correction terms
that are inaccessible in the former. A better analytic grasp of the spectrum of
fermionic conformal field theories can help in many directions including making
new precise analytic predictions for supersymmetric models, computing the
binding energies of fermions in curved space, and describing quantum phase
transitions in condensed matter systems with emergent Lorentz symmetry.
