The paper studies ‘good arrangements’ (transversality properties) of collections of sets in a normed vector space near a given point in their intersection. We target primal (metric and slope) characterizations of transversality properties in the nonlinear setting. The Holder case is given a special attention. Our main objective is not formally extending our earlier results from the Holder to a more general nonlinear setting, but rather to develop a general framework for quantitative analysis of transversality properties. The nonlinearity is just a simple setting, which allows us to unify the existing results on the topic. Unlike the wellstudied subtransversality property, not many characterizations of the other two important properties: semitransversality and transversality have been known even in the linear case. Quantitative relations between nonlinear transversality properties and the corresponding regularity properties of set-valued mappings as well as nonlinear extensions of the new transversality properties of a set-valued mapping to a set in the range space due to Ioffe are also discussed.