This is an announcement for the paper "Continuous multilinear functionals on $C(K)$-spaces are integral" by A. Ibort, P. Linares, and J. G. Llavona. Abstract: In this paper we prove the theorem stated on the title: every continuous multilinear functional on $C(K)$-spaces is integral, or what is the same any polymeasure defined on the product of Borelian $\sigma$-algebras defined on compact sets can be extended to a bounded Borel measure on the compact product space. We provide two different proofs of the same result, each one stressing a different aspect of the various implications of this fact. The first one, valid for compact subsets of $\R^n$, is based on the classical multivariate theory of moments and is a natural extension of the Hausdorff moment problem to multilinear functionals. The second proof relies on a multilinear extension of the decomposition theorem of linear functionals on its positive and negative part which allows us prove a multilinear Riesz Theorem as well. These arguments are valid for arbitrary Hausdorff compact sets. Archive classification: math.FA Mathematics Subject Classification: 46G25 Remarks: 10 pages The source file(s), Integralmultilinear.tex: 39365 bytes, is(are) stored in gzipped form as 0801.2878.gz with size 13kb. The corresponding postcript file has gzipped size 85kb. Submitted from: plinares@mat.ucm.es The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/0801.2878 or http://arXiv.org/abs/0801.2878 or by email in unzipped form by transmitting an empty message with subject line uget 0801.2878 or in gzipped form by using subject line get 0801.2878 to: math@arXiv.org.