What is the difference between be surprised at and be surprised with?

Answer 1

To be 'surprised at' something means to find out something, and react with a feeling of surprise.
To be 'surprised with' something means that someone else surprises someone by giving that person something, doing something for him etc.

Another common expression is to be 'surprised by' something. This means that this something is what surprised you.

Answer 2

I am surprised at your question .... the question surprises me.
He was surprised with a 50th birthday part ... especially since he was only 37.