Use this tag for questions about distributions (or generalized functions). For questions about "probability distributions", use (probability-distributions). For questions about distributions as sub-bundles of a vector bundle, use (differential-geometry).

Distributions are object that generalize the notion of function.
They are linear functionals on a set of *test functions* into the real numbers. The set of test functions is usually $\mathcal{D}(\mathbb R^n)=\mathcal{C}^{\infty}_c(\mathbb R^n)$. The basic idea is to to reinterpret functions as linear functionals acting on a space of test functions.

If we use a larger test space, such as $\mathcal{S}(\mathbb R^n)$ we obtain a smaller space of distributions, called *tempered distributions*. The space of distributions is usually denoted by $\mathcal{D}'(\mathbb R^n)$, while tempered distributions are usually denoted by $\mathcal{S}'(\mathbb R^n)$.

Distributions are heavily used in partial differential equations (when classical solutions don't exist there might still be distributional solutions), physics and engineering.