  This habilitation thesis summarizes the research that I have carried out from
2005 to 2019. It is organized in four chapters. The first three deal with
random planar maps. Chapter 1 is about their metric properties: from a general
map-mobile bijection, we compute the three-point function of quadrangulations,
before discussing the connection with continued fractions. Chapter 2 presents
the slice decomposition, a unified bijective approach that applies notably to
irreducible maps. Chapter 3 concerns the $O(n)$ loop model on planar maps: by a
combinatorial decomposition, we obtain the phase diagram before studying loop
nesting statistics. Chapter 4 deals with random partitions and Schur processes,
from steep domino tilings to fermionic systems.
