We consider the family UREC of unambiguous recognizabletwo-dimensional languages. We prove that there are recognizablelanguages that are inherently ambiguous, that is UREC family is aproper subclass of REC family. The result is obtained by showing anecessary condition for unambiguous recognizable languages.Further UREC family coincides with the class of picture languagesdefined by unambiguous 2OTA and it strictly contains itsdeterministic counterpart. Some closure and non-closure propertiesof UREC are presented. Finally we show that it is undecidablewhether a given tiling system is unambiguous.