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One-argument and two-argument operations of distributional composition are studied for real-valued functions and distributions on R. Several results are proved on the existence of the distributional composition [g] • [f ] for functions (and on the consistency with the usual functional composition g • f ) as well as for distributions, e.g. in case g is a continuous function and f is a measure or, more generally, a semi-regular (almost regular) distribution, i.e. a distribution such that the Łojasiewicz value f Ł (a) of f at a point a exists for almost all a on R and the mapping f R : x → f Ł (x) is a measurable (locally integrable) function. Several formulas involved with the Dirac delta distribution δ are given.

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