Abstract: Toric varieties form a specific class of algebraic varieties equipped with a well-behaved action of an algebraic torus. They provide a useful setting for testing conjectures, as they admit a particularly explicit and combinatorial description. The fundamental theorem of toric geometry states that toric varieties correspond to fans, that is, sets of strongly convex polyhedral cones in $\mathbb{R}^n$ that are closed under taking faces and whose relative interiors are pairwise disjoint. Properties of the fan translate directly into geometric properties of the associated toric variety. In particular, a toric variety is complete if and only if the cones of the fan cover the whole space $\mathbb{R}^n$, and it is non-singular if and only if each cone is generated by part of a basis of the integer lattice $\mathbb{Z}^n$. We focus here on characterizing complete non-singular toric varieties, also called toric manifolds. The Picard number of a toric manifold is the rank of its Picard group; this equals the number of the 1-dimensional cones minus the dimension of its associated fan. There are two major directions of research toward this characterization: studying toric manifolds of fixed (small) dimension, or studying those with fixed (small) Picard number. In dimension 2, toric manifolds are completely understood: they are obtained by a sequence of toric blow-ups startying either from the complex projective plane or from a Hirzebruch surface. In any dimension $n$, the unique toric manifold of Picard number 1 is the complex projective space $\mathbb{C}P^n$, whose fan corresponds to the normal fan of a unimodular $n$-simplex. Kleinschmidt (1988) and Batyrev (1991) classified toric manifolds of Picard number 2 and 3, respectively. In this talk, I will present a sequence of joint works with Suyoung Choi and Hyeontae Jang leading to the classification of toric manifolds of Picard number 4 in terms of fans, relying mainly on a combinatorial construction known as the (simplicial) wedge operation; which was used for instance by F. Santos in his construction for disproving the Hirsch conjecture. I will also discuss recent advances toward the case of Picard number 5.