1. to recognize; to recognise; to observe; to notice
遠くに小さな船の姿を認めた。I spotted a small boat in the distance.
2. to deem; to judge; to assess
3. to approve; to deem acceptable; to allow
とにかく、ちゃんとした理由がないと認められない。Anyway, it won't be approved without a proper reason.
4. to admit; to accept; to confess (to a charge)
あまり認めたくはありませんがね。I don't really want to admit it, though.
5. to watch steadily; to observe carefully
6. to renown; to give renown to; to appreciate; to acknowledge