Regression models with high-dimensional predictors arise in many statistical fields, such as non- and semiparametric regression models based on splines or wavelets and spatio-temporal statistical approaches, but also more generally in bio- and information-technology. In such models the number of parameters to be estimated is typically large compared to the sample size and the resulting inverse problems therefore require some kind of regularisation. In this project, modern Bayesian approaches for regularisation and model choice in regression models with high-dimensional predictors will be developed. These approaches are based on prior distributions which can be employed for both regularisation and model choice, similar in spirit to the popular LASSO criterion. In fact, the LASSO criterion corresponds to one specific prior distribution of this type. Suitable priors with favorable selection and adaptivity properties will be investigated. Inferential procedures based on MCMC simulation techniques will be developed for different types of flexible regression models.