The term handwaving is used in mathematics and physics to describe arguments that are not mathematically rigorous.

Such arguments often include order-of-magnitude estimates and appeals to symmetry or dimensional consistency. Competent, well-intentioned researchers and professors rely on handwaving when, given a limited time, a large result must be shown and minor technical details cannot be given much attention—e.g. "It can be shown that x is even."

The term derives from the use of gestures to add emphasis when stating arguments, and the tendency to continue or increase the gesturing as a substitute for mathematics, when an argument is hard to put across or the proponent is tired of deploying words and/or symbols.

The expression "vigorous handwaving" is meant to suggest that the proponent should be aware of failing to convince; in addition it may suggest that the proponent lacks confidence in the validity of his or her own argument.

Back-of-the-envelope calculations are approximate ways to get an answer by over-simplification and are compatible with handwaving. Handwaving contrasts with use of abstract nonsense, that is, appeal to some intellectual superstructure of generalities.

By extension, handwaving is used in speculative fiction criticism to refer to a plot device (e.g. a scientific discovery, a political development, or rules governing the behavior of a fictional creature) that is left unexplained or sloppily explained because it is convenient to the story, with the implication that the writer is aware of the logical weakness but hopes the reader will not notice. The fictional material "handwavium" (similar to unobtainium) is sometimes referred to in such situations.