To cross over (e.g., a mountain or river); to go across to the other side.
台風で山道を越すのは危険だ。
It's dangerous to cross the mountain road during a typhoon.
国境を越して隣国へ入った。
We crossed the border into the neighboring country.
川を越すための橋が修理中だ。
The bridge to cross the river is under repair.
To get over or overcome (e.g., hardship).
彼は長年の病気を越して復帰した。
He recovered from a long illness and returned.
チームは数々の困難を越して決勝に進んだ。
The team overcame many difficulties and advanced to the finals.
To pass through or get through a period of time (e.g., to get through the winter).
寒さを越すために暖房の準備をする。
We prepare heating to get through the cold season.
この作物は冬を越して春に実を結ぶ。
This crop survives the winter and bears fruit in spring.
To surpass or exceed (in amount, degree); often written with the kanji 超す.
今年の売上は予定を越した（＝超した）。
This year's sales exceeded what was planned.
彼の記録を越すのは簡単ではない。
It's not easy to surpass his record.
To move house; to change residence.
来月、仕事のために東京に越す予定だ。
I'm planning to move to Tokyo next month for work.
彼は大学進学を機に地元から越した。
He moved away from his hometown when he went to university.
Honorific (sonkeigo) for 'to go' or 'to come' (as in お越しになる).
社長が午後二時にお越しになります。
The president will come at 2 p.m.
いつこちらへ越されますか？
When will you come/visit here? (polite)