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The performance of multibranch selection diversity (SD) receivers over L arbitrarily correlated generalised Gamma (GG) fading channels is analysed and evaluated. Following a statistical approach previously proposed for the multivariate Nakagami-m distribution, a closed-form upper bound for the joint GG probability density function (PDF) and an infinite series upper bound for the joint GG cumulative distribution function (CDF) are obtained. For the special case of the trivariate GG distribution (L=3), exact expressions in the form of infinite series for the PDF and CDF are presented. The derived bounds are used to obtain tight upper bounds for the distribution of the SD output signal-to-noise ratio (SNR). Furthermore, based on the derived analysis, novel analytical expressions that accurately approximate the distribution of the SD output SNR are presented. The bounds and approximations are used to study important performance criteria of SD receivers such as outage and average symbol error probability. The accuracy of the derived expressions has been verified, on the one hand, analytically for triple-branch (L=3) and, on the other hand, by means of computer simulations for multibranch (L>3) SD receivers.