To permit or allow — to give permission or approve an action or state.
先生は学生に課外活動への参加を許した。
The teacher allowed the students to participate in extracurricular activities.
両親が夜遅くまで外出することを許してくれた。
My parents let me go out late at night.
会社は出張の申請を許可した（許した）。
The company approved the business trip request.
To forgive, pardon, or excuse — to absolve someone of blame. The kanji 赦す is often used for formal or legal pardons.
彼は長年の恨みを捨て、ようやく彼女を許した。
He finally let go of his long-held grudge and forgave her.
親は子どもの失敗を許して導くことが大切だ。
It's important for parents to forgive their children's mistakes and guide them.
歴史的には、国王が反逆者を赦すことがあった。
Historically, kings sometimes pardoned rebels.
To exempt, remit, or release someone from a penalty, debt, or obligation (to pardon/forgive a burden).
特殊な事情が認められ、罰金の一部が赦された。
Due to special circumstances, part of the fine was remitted.
To acknowledge or concede — to admit or accept a fact or point.
彼の勝利は誰もが認め、もはや異論を許さなかった。
Everyone acknowledged his victory and there was no further dispute.
議論の末、彼女が正しいことを皆が許した。
After the debate, everyone admitted that she was right.
To trust or confide in someone — to let one's guard down and open one's heart.
長い付き合いで、ようやく彼に心を許した。
After a long relationship, I finally trusted him.
彼女は親友にしか心を許さない性格だ。
She only opens up to her close friends.
To give up (points, distance, advantage) — to yield or concede points/lead to the opponent.
試合序盤にミスが重なり、相手に3点を許してしまった。
Early mistakes added up and we conceded three points to the opponent.
彼はレースで先行するライバルにわずかに差を許した。
He gave the leading rival a slight lead in the race.