I have a Panasonic Lumix DMC-TZ3.
I am not a professional photographer, so this is a great 'point and shoot' option for shooting gymnastics.
I researched specifically for cameras that took good photos in low light (all gym meets are low light, since they are indoors), and which still took decent action photos.
It also has a video camera built into it, so I can just turn the dial, and start video-ing, instead of fidgeting around in my bag for a different device, and missing out on action shots while doing so.
The camera also has a NIFTY little function that lets you capture still photos from the videos you took! This is how I handle the action shots: I just video the gym routine. Later, I can go frame by frame, and capture the photos I like.
If you look on my album on this website ("msl529's album"), you will see a couple of photos I captured from video with this camera (the L5 bars tap swing and a couple of beam photos), and you will also see the still photos I got w/ it (family pics, etc.).
Also if you look at the video section of this website, you can view my "L5 State Video" which I took entirely with this little handheld camera. I loaded & edited it on a website called 'One True Media', and then posted it onto You-Tube.
All this can be done by simply removing the memory card from the camera and putting it into your computer (you need a memory card slot in your computer).
The camera is small and easy to carry. I never leave home without it!
I may not get super-professional photos, but I love its ease of use, and do get some very nice pics w/ it.