Classical Hodge theory explains us that the cohomology of a smooth projective complex variety verifies a lot of nice properties. After a few reminders about classical Hodge theory and about tropical varieties, I will present a tropical analogous of these properties for the cohomology of a tropical variety [arXiv:2007.07826]. This tropical Hodge theory is strongly linked to combinatorial Hodge theory, as well as to the limit of families of complex varieties. I will only detail the latter. This will allow me to talk about the tropical Hodge conjecture and to discuss its links with the classical Hodge conjecture. Finally, I will present a work in preparation with O. Amini where we use the tropical Hodge theory to prove the tropical Hodge conjecture for rationally triangulable tropical varieties.
