We present a new definition of distribution semigroups, covering in particular non-densely defined generators. We show that for a closed operator A in a Banach space E the following assertions are equivalent: (a) A generates a distribution semigroup; (b) the convolution operator δ′ ⊗ I − δ ⊗ A has a fundamental solution in D′(L(E, D)) where D denotes the domain of A supplied with the graph norm and I denotes the inclusion D → E; (c) A generates a local integrated semigroup. We also show that every generator of a distribution semigroup generates a regularized semigroup. Introduction Distribution semigroups in Banach spaces and their generators were introduced by J. L. Lions in [16] as a generalization of C0-semigroups and their generators. Like the generators of C0-semigroups the generators of the distribution semigroups in [16] are densely defined. It was then shown in [16] that the densely defined closed linear operators A that generate a distribution semigroup are exactly those for which the convolution operator δ′ ⊗ I − δ ⊗ A where I denotes the inclusion D(A) → E has a fundamental solution. Treating Cauchy problems as convolution equations has turned out to be very useful (see [6], [12], [13] and [14]). But in doing so the assumption that D(A) is dense in E is unnecessary in most of the results. On the other hand in the last years there has been an interest in dealing with operators that are not densely defined, starting perhaps with [7]. Reasons for this interest may be that it might not be so easy to decide whether a given operator is densely defined and the difficulties that arise when passing from E to E∗ or l∞(E). So the question how the theory of distribution semigroups introduced in [16] can be extended to cover non-densely defined generators in such a way that the above characterization via fundamental solutions holds seems natural. It turns out that this indeed is possible in a way that even simplifies the original definitions given in [16]. Moreover, the local strongly continuous representations (see section 4) of the distribution semigroups thus obtained are exactly the local integrated semigroups (for the distribution semigroups in the sense of [16] a relation to local integrated semigroups has been obtained in [23], Theorem 5.6; see also [15], Theorem 7.6 or [2], Theorem 7.2). So the theory presented here Received by the editors October 17, 1995 and, in revised form, February 6, 1997. 1991 Mathematics Subject Classification. Primary 47D03, 34G10, 47A10, 46F10. c ©1999 American Mathematical Society