The general theory of Shimura varieties has been developed in the 1970?s by Deligne. It generalizes classical objects such as modular curves or Hilbert-Blumenthal varieties. Such varieties naturally appear in the search for higher reciprocity laws as studied within the Langlands program. The arithmetic properties of these varieties, which are encoded in their reduction to positive characteristic, have contributed to some of the most spectacular developments in arithmetic geometry in the last decades. In my talk I concentrate on the geometry of these objects and survey some recent results in the reduction to positive characteristic.