Flash Compatibility in Web Browsers?

I was wondering, because of my recent liking of AS3 and active flash content:

Just how compatible are flash files in web browsers?

Will they run in every browser? (obviously not, but which ones don’t they run in / which ones do they have trouble running in?)

Why, in Internet Explorer does it always give me a hard time about running active content with its security warnings? (is there a way to get around this / allow the content to play normally without hassle? e.g When I got to Sony.com Internet Explorer does not complain about running that active content.)

Is there anything else you know about the compatibility that you would like to share?