Tutorial: Multimeters – Understanding the functions of a multimeter @ The Adafruit Learning System.

If there is not enough current, your circuit may not be able to do the work it was designed to do.  Logic circuits may not function reliably, displays may be dim, motors may stall. On the other hand, if there is too much current, things will heat up and components may be damaged. In extreme cases there may even be smoke or flames.

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