Indicates means or instrument; often used as をもって meaning 'by' or 'by means of'.
この書類をもって入場を許可します。
Admission will be permitted with this document.
証拠をもって反論したが、受け入れられなかった。
I rebutted with evidence, but it was not accepted.
Indicates cause or reason; often in をもって meaning 'because of' or 'due to' (formal/written).
人手不足をもって、本日の業務は中止となりました。
Due to a lack of staff, today's operations have been canceled.
諸般の事情をもって延期することになりました。
It has been decided to postpone it because of various circumstances.
Marks a point in time (as of, effective from); used in formal phrases like 'as of today' (本日をもって).
本日をもって当店は閉店いたしました。
As of today, this shop has closed.
10月1日をもって新しい料金が適用されます。
The new rates will apply as of October 1.
Used occasionally to add emphasis to the preceding statement (somewhat literary/formal).
彼の研究は深く、もって学界の評価も高い。
His research is profound, and thus he is highly regarded in academia.
Used (often as でもって) as a conjunction meaning 'and furthermore' or 'in addition' (colloquial).
彼は英語が話せる、でもって料理も得意だ。
He can speak English, and moreover he is good at cooking.
Used to mean 'therefore' or 'hence' to draw a conclusion (formal/old-fashioned).
彼は重い病を負っている、もって出席は叶いませんでした。
He was suffering from a severe illness; therefore he was unable to attend.