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Jerusalem
Yes, they do. See the related question on why that might be.
Both claim the entire Southwest Levant, termed by the Jews as "Eretz Yisrael - The Land of Israel" and by the Arabs as "Belaad Filastin - The Country of Palestine", as their homeland.
A discussion and explanation of Jesus claim that if you believe in him you will never die.
Historic Palestine or the Land of Israel. (Both are regional names for the same piece of land, roughly.)
It depends on how they make that claim. Bedouin and Druze claim that Israel belongs to them as much as it belongs to the Jews and proudly call themselves Bedouin Israelis and Druze Israelis. A number of Arabs who formerly lived in what is now Israel or the Occupied Territories claim that the Jewish State is illegitimate and demand that country be returned to them for administration. These individuals are called Palestinians. (Admittedly, a number of Palestinians are willing to allow Israel to exist outside of the Occupied Territories, but there is no difference in naming between the Two-Staters and the One-Staters.)
Islamic
Muslim
the Turks and the Greeks are disputing claim over Cyprus
Arabs believe they have a right to claim land for exactly the same reasons that all other people believe that they have a right to claim land. They feel a right to the land that they live on since they live there and their ancestors have lived there.
The Old City of Jerusalem is known for its sacred sites in the Jewish, Muslim, and Christian faith, solidifying each group's historical claim to the land. Both the Palestinians and Israelis want Jerusalem as their capital.
It depends on the instance. The Nazis made the claim that anyone who had one Jewish grandparent was Jewish. The Israelis have slightly different criteria. The Spanish Inquisition made the claim that someone was a Jew if that person claimed to be a Jew or if they failed to be a "proper" Christian.