Through the lens of Minimal Model Program, the classification of algebraic varieties can be summarised in 2 steps. First MMP allows to decompose a variety with mild singularities, birationally, into a tower of fibrations whose general fibres have ample, anti-ample or numerically trivial canonical divisor. The natural second step is to study these 3 classes, e.g. by their moduli theory or other properties that shed light on all elements in the classes. It turns out a very important property is boundedness: a collection of varieties is bounded when the elements of the given collection can be parametrised using a finite type geometric space. This is crucial in the construction of proper moduli spaces of finite type. Moreover if a given collection of algebraic varieties is bounded (in char 0) then the topological types of their underlying analytic spaces belong to finitely many homeomorphism classes and their topological invariants come in finitely many versions. While over the past 15 years several breakthroughs have completely settled the question of boundedness (and subsequent construction of moduli spaces) in the case of log canonical models (varieties/pairs with ample canonical divisor) and Fano varieties (anti-ample case), the situation is still quite unclear in the case of trivial numerical divisor. I will try to explain what is known or not, and which challenges make the situation quite more complicated than the other cases. I will explain how we can overcome most issues if we assume that a K-trivial variety is endowed with a fibration structure of relative dimenesion one. The seminar is based on various works I developed over the past 10 years with G. Di Cerbo, C. Birkar, S. Filipazzi and C. Hacon. Time permitting I will talk about work in progress with P. Engel, S. Filipazzi, F. Greer, M. Mauri showing various new boundedness results for K-trvial varieties fibered in K3 surfaces or abelian varieties