Historically, the main impediment to more widespread employment of Bayesian methods has been the difficulty of performing required calculations as they require high dimensional numerical integrations. In the past decade, major advances have been made in resolving this problem so that previously inaccessible problems can be readily carried out This book is devoted to describing those advances. It incorporates the various techniques which have been developed and gives examples. Several case studies are used to illustrate the implementation of a fully Bayesian data analysis employing these new techniques. This volume comprises a unique collection of material, summarizing a fast moving area.