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Every Mac has at least one chip that stores certain small bits of information even when the power is off. Depending on your Mac model, it may have a PMU (power management unit), SMU (system management unit), SMC (system management controller), NVRAM (non-volatile RAM), PRAM (parameter RAM), or some combination of these. Although it's uncommon, any of this memory can become corrupted, leading to symptoms including problems turning on, sleeping, or waking up; excessive fan noise; not displaying video; and, for portables, not recognizing or charging a battery.
In the event that you encounter such symptoms, I'd like to tell you how to reset these various sorts of memory, but the instructions vary significantly from one Mac model to the next. Also, in some cases a single technique can reset all the relevant chips, while in other cases you may have to follow one procedure for, say, the SMU and another to reset the PRAM. So instead of spending several pages laying out all the details, I've listed the pages on Apple's Web sites that describe the reset procedures for each product line: