A wave is a disturbance that propagates through space and time, usually with transferrance of energy. While a mechanical wave exists in a medium (which on deformation is capable of producing elastic restoring forces), waves of electromagnetic radiation (and probably gravitational radiation) can travel through vacuum*, that is, without a medium.
*A vacuum is a volume of space that is essentially empty of matter, such that its gaseous pressure is much less than standard atmospheric pressure.
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