When the Assyrian king Shalmaneser conquered the northern kingdom of Israel in 722 B.C., he deported its ten tribes as captives to Assyria. Thereafter, they completely disappeared from world history.
Today's Jews, therefore, are primarily descended from the southern kingdom's remaining two tribes.
The Assyrian empire collapsed around the year 613 B.C. when its capitol city Nineveh was taken (even though it had been considered unconquerable). Assyria was then incorporated into the Babylonian empire.