Real-valued differential forms on Berkovich analytic spaces and their cohomology

We study superforms on tropical varieties and differential forms on Berkovich spaces. For superforms on tropical varieties, we extend the previous defintionts, prove a Poincare lemma and prove Poincare duality for tropical manifolds. For differential forms on Berkovich spaces we also prove a Poincare lemma and finite dimensionality results. We further define new, but equivalent, approaches to differential forms on Berkovich spaces.