This PhD thesis contributes to the modeling of overdispered count data in three ways. First, we extend two approaches for building robust M-estimators of the regression parameters in the class of generalized linear models to the negative binomial (NB) distribution. Second, we adapt recently developed tests, such as the so-called saddlepoint test, to the framework of overdispersed count data and give a detailed account of their computation and implementation. Through extensive simulations we compare them to traditional tests in order to assess their effective level under the null hypothesis and their power under alternatives, and this for models based on a full NB likelihood or on moment restrictions only. Finally, we enlarge our scope to the class of mixed compound Poisson models, where the overdispersion is due to the variance of unobserved mixing variables. We propose a semi-parametric framework where the mixing distribution is left unspecified and estimated by point-masses.