This is an announcement for the paper "Random series of trace class operators" by Gilles Pisier. Abstract: In this lecture, we present some results on Gaussian (or Rademacher) random series of trace class operators, mainly due jointly with F. Lust-Piquard. We will emphasize the probabilistic reformulation of these results, as well as the open problems suggested by them. We start by a brief survey of what is known about the problem of characterizing a.s. convergent (Gaussian or Rademacher) series of random vectors in a Banach space. The main result presented here is that for certain pairs of Banach spaces $E,F$ that include Hilbert spaces (and type 2 spaces with the analytic UMD property), we have $$ R(E\widehat\otimes F) =R(E)\widehat\otimes F + E\widehat\otimes R(F) $$ where $R(E)$ denotes the space of convergent Rademacher series with coefficients in $E$ and $E\widehat\otimes F$ denotes the projective tensor product. Archive classification: math.FA math.OA math.PR Mathematics Subject Classification: 46B09 Citation: Proceedings Cuarto CLAPEM Mexico 1990. Contribuciones en The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1103.2090 or http://arXiv.org/abs/1103.2090