My device is getting slow, apps start misbehaving/crashing. What can I do?

Everything was working fine for a long time. But suddenly the device is getting slower than usual (e.g. apps react slowly when starting an action -- like the RSS reader takes "ages" to open an article). Also, several apps start crashing: force-closes as well as unexpected restarts. Sometimes this even involves some automatic reboot of the device. Is there anything I can do, besides doing a factory-reset? Preferably it should work on un-rooted devices. Remark: not all described problems may occur together, so on one device e.g. there might be no automatic reboot involved. Question remains the same, nonetheless.