To challenge someone (to a fight, match, etc.) — to throw down the gauntlet or contend directly with another party.
若手選手が reigning champion に挑んだ。
The young athlete challenged the reigning champion.
彼は相手チームに直接挑んで試合を決めた。
He directly challenged the opposing team and decided the match.
彼女は社内のリーダーの座に挑むと表明した。
She announced she would challenge for the leadership position within the company.
To tackle or attempt something — to take on a problem, goal, or record; to go after an objective.
難しい研究課題に挑むには計画が必要だ。
You need a plan to tackle a difficult research topic.
彼は自己ベスト更新に挑んでいる。
He is attempting to beat his personal best.
若者たちが起業に挑んでいる。
Young people are trying their hand at starting businesses.
To make sexual advances — to pressure someone for sex or make overt advances (can be explicit or coarse).
被害者は飲み会で同僚にしつこく挑まれたと訴えた。
The victim reported being persistently propositioned by a colleague at a drinking party.
相手の意思を無視して挑むのは許されない。
It is unacceptable to make advances while ignoring the other person's wishes.