A noun (often used as a suffix) meaning to leave something entirely to someone or something; to entrust/resign responsibility to another.
全部彼任せにするのは良くない。
It's not good to leave everything up to him.
成り行き任せでプロジェクトを進めるわけにはいかない。
We can't run the project leaving things to chance.
説明は任せろ、私が対応するから。
Leave the explanation to me — I'll handle it.