잘 부탁해 (casual) · 잘 부탁해요 (polite) · 잘 부탁드립니다 (formal)
I look forward to working with you; please treat me well
Example
처음 뵙겠습니다. 앞으로 잘 부탁드립니다.
Nice to meet you. I look forward to working with you.
Word parts
부 hand over+탁 entrust
Literally 'I ask you a favor well' — asking for goodwill in a new relationship.