  In this review the physics of Pfaffian paired states, in the context of
fractional quantum Hall effect, is discussed using field-theoretical
approaches. The Pfaffian states are prime examples of topological ($p$-wave)
Cooper pairing and are characterized by non-Abelian statistics of their
quasiparticles. Here we focus on conditions for their realization and
competition among them at half-integer filling factors. Using the Dirac
composite fermion description, in the presence of a mass term, we study the
influence of Landau level mixing in selecting a particular Pfaffian state.
While Pfaffian and anti-Pfaffian are selected when Landau level mixing is not
strong, and can be taken into account perturbatively, the PH Pfaffian state
requires non-perturbative inclusion of at least two Landau levels. Our
findings, for small Landau level mixing, are in accordance with numerical
investigations in the literature, and call for a non-perturbative approach in
the search for PH Pfaffian correlations. We demonstrated that a method based on
the Chern-Simons field-theoretical approach can be used to generate
characteristic interaction pseudo-potentials for Pfaffian paired states.
