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.
