Issues of this sort are, unfortunately, more common and the we'd like them to be. However, they tend not to be cause by Firefox itself, but rather by software that is running either inside or alongside it, such as add-ons or firewalls. Since these issues have a broad span of possible causes, you need to diagnose your problems before fixing them. Follow these instructions to the line, as they'll probably help you diagnose and fix the problems you're having.

1. run Firefox in safe-mode to disable all extensions, themes and plugins. If this fixes your problem, be it with RAM or CPU usage, then you know it's a problem with add-ons (plugins, themes or extensions). Proceed to number 2. If safe-mode doesn't fix your problem, then read bellow, after this list;
2. update all extensions (go into Tools > Add-ons > Extensions > Find Updates), themes and plugins (see this page in your Firefox. If this doesn't solve the issues, proceed to the following number;
3. disable all extensions, themes and plugins in your Firefox (not running safe-mode). This is temporary and for diagnostic purposes only. Being certain that, as in safe-mode, the problems you're having have gone away, enable one plugin at a time. You should be certain that you actually want that plugin to be enabled. It's best to keep the overall number of enabled plugins as low as possible. When you encounter problems, you know you've found a problematic plugin, so disable it for good. Keep enabling all plugins (except problematic ones) until you've gone through them all. If you need support for a problematic plugin, you need to contact its author for support;
4. enable one extension at a time. Again, be certain that you actually want that extension to be enabled. As with plguins, it's best to keep the overall number of enabled extensions as low as possible. Also, try the theme you want to have installed so see if that is what's causing the problem. When you encounter problems, you know you've found a problematic extension/theme, so disable it for good. Keep enabling all your extensions (except problematic ones) until you've gone through them all. Just the same as with plugins, if you need support for a problematic extension/plugin, you need to contact its author;
5. if you've followed my instructions above, you're done! You've fixed your problems with problematic add-ons. If you want to keep using those problematic add-ons, please contant their authors for support.

Ok, now... If disabling all extensions and plugins through safe-mode didn't work to fix your problems (or, in other words, if you've just read number 1 on the list above and come straight here), then you have different issue. The most likely scenario is that you have a third party software running on your computer that is messing with Firefox. Detecting which program it is may be tricky, but the following list should help you. Make sure you follow it carefully. Don't forget to answer the question on the last point, if nothing else helps.

1. try reinstalling Firefox. No data will be lost. You can get the latest version (for free, as always) at getfirefox.com. Make sure you uninstall Firefox prior to reinstalling it. For help installing Firefox, see this support article. If that doesn't fix the problem, proceed;
2. do a virus/malware check on your computer. See this support article for help on this point. This is a very important step, so please pay attention to it. If your problem is not due to viruses/malware, proceed;
3. disable all software running in the background that you don't want to have running in the background (in Windows operating systems, this is done by pressing WINDOWS+R in your keyboard, typing msconfig and pressing Enter; now, under the Startup tab, you can uncheck the software you don't want, and reboot your system for changes to take effect; if you're unsure of what software you want running, ask someone with more experience). If this doesn't fix your issues with Firefox, proceed;
4. check if your firewall/antivirus/security suite is conflicting in any way with Firefox's normal behavior. Check for enabled functions/features that you don't want and/or may be causing problems with Firefox. You'll find that these features are most likely tied to Internet Security features, such as link scanners or URL checkers and the like. If you're not sure they are conflicting with Firefox, simply try to disable them temporarily to see whether or not that's true. If this doesn't solve the issues, proceed to the following number;
5. check your operating system security options, mainly advanced options that are not configured by default. While it's very unlikely that this may be the cause of the problem, it's remotely possible. If this doesn't work, proceed to the following point;
6. update your modem/router software. There have been some reports that some modem/router software may cause Firefox and other browsers to loose performance and/or stability. If updating doesn't fix your issue, try other versions of the software, if possible. If you need support with this, contact your modem/router manufacturer. If this doesn't help, proceed;
7. if you are using a Windows operating system, clean up your OS registry using appropriate software. There have been reports that badly maintained Windows Registries may case problems with Firefox. If this doesn't help, see the following point;
8. please try creating a new temporary Firefox profile (managing profiles) and see if the issues persist. Please report the results, so we can help you further.

That reply above is cool...but too much txt for me (srry). Safe mode is definately a good place to start and if ok there...then find out which add-on to blame... I have a couple of candidates :-

AVG Toolbar (I'm sure this has come up before) and ( a new one) Adobe DLM (sorry, its gone - think it was a download manager from Adobe)

Thinsk we're ok now. Might try Crome one day.

Why on Earth does Mozilla support think that "there...then" is a URL ? Odd.

Solved my problem. I had installed a Persona (skin) that used animations. This seems to have been the problem.

What I did not appreciate is that when I disabled the Personas add-on, it did nothing to stop the behavior, and left the Persona intact.

After switching to another non-animated Persona, my cpu is back to 0%

I discovered this after lengthy experimentation with the above suggestions, which did nothing for me.

i have the same issue, playing flash etc only builds more cpu use as plugin container requests 20 or more percent cpu use on top of FF's 100percent all the time even while idle.. it doesnt seem to decrease or increase with activity..

to solve this in xp i reduced firefox.exe's it's priority to low.. it still requets 100% cpu use about 90% of the time, but i can play flash and multitask like i have in previous versions...

i also had to overclock my nvidia 5500 softquadroed 500/600

to 333/430 to help compute and render videos smoothly..

only then did it start to use ram like it should..

something is causeing memory to not be used properly, and massive loops in codeing or something to make the cpu work harder then it needs too...

please fix this soon. it could crash pc's or even burn things out.. ram and cpu could easily overheat.. even when it seems it's not using it, it's addressing and dropping that qued memory agressivly.. and this is causeing excessive heat and use. which could actually damage people's pc's.

I disabled the add-on "Microsoft .NET Framework Assistant 1.1" and my problems went away.