Byzantines inhereted fallen Roman Empire, or to be more precise, Roman Empire first splitted in Western and Eastern, Western fell Eastern stayed.. later when Ottomans have destroyed Eastern Roman Empire, Zoe(or Sophia she was called in west), niece of of the last Eastern Roman emperor Constantine XI, married with Ivan III of Russia, so the blood line of Roman Empire is transmited to Russian royal family and they considered themselves as inheritors of Roman Empire.. but do you buy into it or not depends on your view point.

You can play connect the dots any way you want, as there really isn't any such thing as an 'official' successor state. Often, there are multiple claims to being the 'true' successor.

The Holy Roman Empire regarded itself as a Roman successor state, and the Austrian Empire could be viewed as a successor to it through the continuity of Hapsburg rule and territorial overlap. And then Austro-Hungary and then just Austria. So why not the President of Austria?