  The main objects under consideration in this thesis are called maps, a
certain class of graphs embedded on surfaces. Our problems have a powerful
relatively recent tool in common, the so-called topological recursion (TR)
introduced by Chekhov, Eynard and Orantin.
  We call a map fully simple if it has non self-intersecting disjoint
boundaries, and ordinary if such a restriction is not imposed. We study the
combinatorial relation between fully simple and ordinary maps with the topology
of a disk or a cylinder, which reproduces relations between moments and free
cumulants established in the context of free probability. We propose a
combinatorial interpretation of the exchange symplectic transformation of TR.
We provide a matrix model interpretation for fully simple maps via the formal
hermitian matrix model with external field and deduce a universal relation
between generating series of fully simple and ordinary maps, which involves
double monotone Hurwitz numbers. In particular, we obtain an ELSV-like formula
for double $2$-orbifold strictly monotone Hurwitz numbers.
  We consider ordinary maps endowed with an $O(\mathsf{n})$ loop model, which
is a classical model in statistical physics, and determine which shapes are
more likely to occur by looking at the nesting properties of the loops
decorating the maps. We want to study the limiting objects when the number of
vertices becomes arbitrarily large, which can be done by studying the
generating series at dominant singularities. We analyze the nesting statistics
in the $O(\mathsf{n})$ loop model on random maps of arbitrary topologies in the
presence of large and small boundaries, relying on previous results for disks
and cylinders and TR for this model. We study the generating series of maps
which realize a fixed nesting graph and characterize their critical behavior in
the dense and dilute phases.
