In this paper, performance of a device-to-device (D2D) communication system is analyzed over double-generalized Gamma (dGG) fading channels. The dGG is a generic distribution for modeling double-scattering fading conditions. Co-channel interference (CCI) caused by various wireless devices in the system is also considered. The CCI fading channel is assumed to be Nakagami distributed. Analytical expressions for important statistical metrics, i.e. probability density function (PDF) and cumulative distribution function (CDF) of signal-to-interference ratio (SIR), are presented. Based on these statistical parameters, expressions for the outage probability, channel capacity and symbol error rate (SER) of the D2D communication system are presented. The performance of D2D system is then discussed and analyzed with the help of numerical results with arbitrary channel fading, path-loss and interference conditions.