General:
* Meetings
* Interactions
* Engagements
* Confrontations
* Brushes
* Run-ins
* Crossings
* Exchanges
* Experiences
* Occurrences
More specific:
* For a brief, unplanned meeting:
* Bump-ins
* Chance meetings
* Casual encounters
* For a hostile or difficult meeting:
* Clashes
* Showdowns
* Standoffs
* For a romantic meeting:
* Rendezvous
* Trysts
* For a spiritual or philosophical meeting:
* Epiphanies
* Revelations
* For a literary or artistic meeting:
* Interludes
* Episodes
Informal:
* Runs into
* Comes across
* Meets up with
The best synonym will depend on the specific situation and the tone you want to convey.