In this thesis, we propose novel Bayesian Nonparametric hypothesis testing procedures for correlated data. First, we develop and study a proposal for comparing the distributions of paired samples. Next, we propose and analyze a hypothesis testing procedure for longitudinal data analysis. Both proposals are based on a flexible model for the joint distribution of the observations. The flexibility is given by a mixture of Dirichlet processes. Besides, for setting up the hypothesis testing procedures, we use a hierarchical representation with a spike-slab prior specification for the base measure of the Dirichlet process and a prior specification on the space of models. For the paired sample test, we use an appropriate parametrization for the kernel of the mixture to facilitate the comparisons and posterior inference. Consequently, the joint model allows us to derive the marginal distributions and test whether they differ or not. The procedure exploits the correlation between samples, relaxes the parametric assumptions, and detects possible differences throughout the entire distributions. For the longitudinal data, we propose to use a mixture of Dependent Dirichlet Processes to capture the correlation between the repeated measurements. The weights of the mixture are built via a stick-breaking prior, that comes from a Markovian process evolving in time. The effect of the predictors is modeled by the underlying atoms. The proposal can provide an estimation of the density through the time for different levels of the predictors, and at the same time can identify the effect of the predictors, without assuming restrictive distributional assumptions. We show the performance throughout the document of our proposals in illustrations with simulated and real data sets. Finally, we provide concluding remarks and discuss open problems.