The intertestamental period refers to the history of the Land of Israel in the 400 years between the Old Testament and the New Testament. Some of the Deuterocannonical books were written during this period.
539 BC–333 BC, Persian rule. For two hundred years after the Babylonian captivity, Judea remained under Persian rule. However, Cyrus the Great allowed the Jewish people to return home, rebuilding Jerusalem and the Temple. The antipathy between Jews and Samaritans also began during this period.
The Hebrew Bible (Old Testament, minus the Deutercannon), ends about halfway through the Persian period.
Egyptian rule 323 BC–198 BC. Syrian rule: 198 BC–165 BC. After the death of Alexander, his kingdom was divided between his generals. Israel lay on the border between Egyptian and Syrian territory, leading to a struggle between the Ptolemies of Egypt and the Seleucids of Syria. The Seleucid rule was a dark period in Jewish history. The last Syrian king, Antiochus Epiphanes, committed many outrages against the Jews, sought to establish idolatry in Jerusalem, and defiled the temple.
Maccabees rule, 165 BC–63 BC. Judas Maccabeus revolted against Syrian rule and won Jewish independence in 166 BC. The priest Matthias and his sons defeated the Syrians in a series of battles. Israel was ruled by the Maccabees and their decendants, the Hasmonean dynasty.
The Roman general Pompey conqured Jerusalem in 63 B.C. The Romans established a large stable government, uniform laws, and good roads. Palestine was ruled by a series of puppet kings installed by Rome, including Herod the Great (37 BC to 4 BC).