•Bayesian forecasting is adopted to develop probabilistic solar power forecasts.•Copulas are derived to model the dependencies between solar power and temperature.•The method is comprehensively examined by various models, locations, and times.•Bayesian forecasting outperforms benchmarks in 55 out of total 64 testing cases.•The accuracy improvement is 3.72 With increased penetration of solar energy sources, solar power forecasting has become more crucial and challenging. This paper proposes a copula-based Bayesian approach to improve probabilistic solar power forecasting by capturing the joint distribution between solar power and ambient temperature. A prior forecast distribution is first obtained using different underlying point forecasting models. Parametric and empirical copulas of solar power and temperature are then developed to update the prior distribution to the posterior forecast distribution. A public solar power database is used to demonstrate effectiveness of the proposed method. Numerical results show that the copula-based Bayesian method outperforms the forecasting method that directly uses temperature as a feature. The Bayesian method is also compared with persistent models and show improved performance. This article includes supplementary material (data and code) for reproducibility.