First: Assyrian (when the Assyrians conquered Israel, the Hebrew inhabitants were scattered all over the Middle East);
Second: Babylonian (when Nebuchadnezzar deported the Judaeans, he allowed them to remain in a community);
Third: Greek (the Jews retained their religion; both under the Persians and the Greeks. The Jews despised the Greeks);
Fourth: Roman (the Romans destroyed Jerusalem, annexed Judaea as a Roman province, and systematically drove the Jews from Palestine).
Diaspora is the scattering of a people from their homeland (referred to the people of that dispersion and describe the experience of the Jewish people). Jewish communities exist in much of the ancient world of the Mediterranean and the Near East (including Babylon, Egypt, Greece, and Rome). This question is about the spread of Judaism.