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The product of two zero mean correlated normal random variables, and moregenerally the sum of independent copies of such random variables, has receivedmuch attention in the statistics literature and appears in many applicationareas. However, many important distributional properties are yet to berecorded. This review paper fills this gap by providing the basicdistributional theory for the sum of independent copies of the product of twozero mean correlated normal random variables. Properties covered includeprobability and cumulative distribution functions, generating functions,moments and cumulants, mode and median, Stein characterisations,representations in terms of other random variables, and a list of relateddistributions. We also review how the product of two zero mean correlatednormal random variables arises naturally as a limiting distribution, with anexample given for the distributional approximation of double Wiener-It\^{o}integrals.

The basic distributional theory for the product of zero mean correlated normal random variables