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Gaussian class multivariate distributions with single integral representations were recently proposed to obtain correlated fading channels with generalized correlation. In this letter, we focus on correlated generalized Rician fading channels, derived from Gaussian class multivariate distributions. We present a comprehensive investigation on the statistical properties of correlated generalized Rician fading channels. Specifically, closed-form expressions for joint probability density function (PDF), marginal PDF, power correlation, joint moment, and joint characteristic function are elaborated. These statistical characterizations help us have a better understanding of the correlated generalized Rician channels and allow for some new applications in performance analysis of diversity schemes and, particularly, maximal ratio combining over correlated fading channels with generalized correlation.