• CD space
• measured Gromov Hausdorff convergence
• non-branching
• optimal transport

Since the first works on CD spaces, it has been clear that the non-branching assumption, associated with the CD condition, could confer some nice properties to a metric measure space, such as the tensorization property, the local-to-global property and the existence of a Monge map. The relation between non-branching assumption and CD condition was made even more interesting by the work of Rajala and Sturm. They proved that the strong CD condition implies a weak version of the non-branching one, that they called essentially non-branching. These results are the main motivation for this master thesis, whose aim is to investigate different aspects of the relation between CD conditions and non-branching assumptions. In particular I explain the main properties of non-branching CD spaces and analyze the stability of non-branching conditions in the context of CD spaces. I exhibit an example of an highly branching CD space, that shows how a weak curvature dimension bound is not sufficient to deduce any type of non-branching condition. Then I show a purely metric stability statement, for the so called very strict CD condition, that does not require any particular analytic assumption. As an application I prove the very strict CD condition for the metric measure space R^N, equipped with a crystalline norm and with the Lebesgue measure.