Title: Some remarks on geometry of quaternionic Shimura varieties

Abstract: Quaternionic Shimura varieties are quotients of the n-fold product of the complex upper half plane by arithmetic groups obtained from orders in quaternion algebras over number fields. They may be considered as generalisations of the Hilbert modular varieties. 

In my talk I intend to give an introduction in the construction of quaternionic Shimura varieties and to discuss some questions about their geometry, such as the question on the Kodaira dimension and also the question on the fundamental group.