It’s taken various forms over the years. The name means “void,” which implies their purpose: that until silence and emptiness is reached, the world will never know peace.
Naturally, silence and emptiness seems to mean complete genocide for everybody.
The Tohu, however, has approached this goal consistently through one subtle but effective method: by focusing on “purity. It can be disturbingly easy to turn people groups against one another based on the erroneous belief that one lineage/blood type/magic/religion/race/People is better than another.