When the variable that is being tested has an unknown distribution.
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Yes, Chis squared test are among the most common nonparametric statistics tests.
A classic would be the Kolmogorov-Smirnov test.
If the distribution is parametric then yes.
Fisher's exact probability test, chi-square test for independence, Kolmogorov-Smirnov test, Spearman's Rank correlation and many, many more.
A nonparametric classifier is a kind of classifier that can work with unknown density function of the classes of a dataset.