Let me start the (hopefully long) list of answers by addressing the USA:
- NSF: (a) Funding & Awards page, (b) Funding Search (division of mathematical sciences)
- AMS: AMS programs page, which for example leads to (a) travel grants, (b) many grants, prizes, and other opportunities, (c) several fellowships.
This answer is CW, feel free to edit it to include more information.