Are there any free/open source alternatives to Adobe After Effects?

Try Blender. It's a free, open source content creation suite that has an impressive number of features:

• Two built-in rendering engines, Blender Internal (fast) and Cycles (realistic), along with other experimental engines such as Freestyle
• A game engine
• The Bullet physics engine, which can be integrated with the above
• A text editor
• An image editor
• A non-linear editor that works with both audio and video
• Tight integration with the open-source, highly popular Python programming language:
  • Python syntax highlighting in the text editor
  • An interactive Python prompt
  • A Python-scriptable interface accelerated with OpenGL
  • Extension with Python plugins
  • Python integration in the game engine
• Motion tracking
• Collada import/export
• Compositing
• A node editor for Blender highly flexible Cycles rendering engine
• CUDA/OpenCL-based GPU rendering for Nvidia cards
• Fluid, cloth, smoke, and fire physics simulations

And much, much more. Not to mention a rapid release cycle (~2 months), a large amount of high-quality documentation (both free and paid; see Blender Cookie), a large, vibrant community, a Blender SE proposal on Area51 currently in the commit stage and four successful Creative-Commons-licensed movie projects.

How does this apply to you? Using the aforementioned motion tracking feature allows you to utilize Blender's power to generate some pretty good special effects (not quite on par with After Effects - it is primarily a 3D modeling application, after all). As Blender's motion tracking algorithms get improved and the visual effects get more realistic, using Blender as a replacement for After Effects should become an easier and easier switch to make.

Having used after effects a fair bit in the past and had a good search around on the open source community the simple answer is unfortunately no.

There are plenty of non linear editors, which are like adobe premier, but nothing really like after effects. The closest I can think of would be kdenlive which I have heard has a few effects (although I've never used it), however I don't think any of them really stand up to after effects.

TBH I unfortunately switch back to my windows partition for doing any serious video editing like that because After effects doesn't work well under wine either.

I would absolutely love to be proved wrong and someone know a really good effects program, but I suspect for the moment I'm going to be dissapointed.

For anyone coming here through Google searches, a lot has changed since this post was made.

Firstly, Blender has gotten much better at compositing and effects. There have been some really excellent uses of this, a (slightly outdated) example of which being Tears of Steel.

Secondly, the software Jahshaka has been mentioned a couple times in this thread. It was dismissed due to being out of date. Well, as of 2013, it has developers working on it again, although it seems that their progress is extraordinarily slow.

There are, of course, other changes, but I saw those two as being the most notable. Please correct me if I missed something significant concerning the programs that have already been mentioned in this thread.