We propose two multivariate extensions of the Bayesian group lasso for variable selection and estimation for data with high dimensional predictors and multidimensional response variables. The methods utilise spike and slab priors to yield solutions which are sparse at either a group level or both a group and individual feature level. The incorporation of group structure in a predictor matrix is a key factor in obtaining better estimators and identifying associations between multiple responses and predictors. The approach is suited to many biological studies where the response is multivariate and each predictor is embedded in some biological grouping structure such as gene pathways or brain regions. Our Bayesian models are connected with penalized regression, and we prove both oracle and asymptotic distribution properties under an orthogonal design. We derive ecient Gibbs sampling algorithms for our models and provide the implementation in a comprehensive R package called MBSGS available on the CRAN. The performance of the proposed approaches is compared to state-of-the-art variable selection stratégies on simulation data set.