To entrust something to someone; to leave (a matter) in someone's hands. A variant of 任せる, sometimes somewhat literary; often appears in the passive.
重要な案件は部長に任すことにした。
I decided to leave the important matter to the manager.
その仕事は彼に任せておけば安心だ。
It's safe to leave that job to him.
交渉は若い社員に任された。
The negotiations were entrusted to a young employee.