We examine the problem of modeling and forecasting European day-ahead and month-ahead natural gas prices. For this, we propose two distinct probabilistic models that can be utilized in risk and portfolio management. We use daily pricing data ranging from 2011 to 2020. Extensive descriptive data analysis shows that both time series feature heavy tails and conditional heteroscedasticity and show asymmetric behavior in their differences. We propose state-space time series models under skewed, heavy-tailed distributions to capture all stylized facts of the data. They include the impact of autocorrelation, seasonality, risk premia, temperature, storage levels, the price of European Emission Allowances, and related fuel prices of oil, coal, and electricity. We provide rigorous model diagnostics and interpret all model components in detail. Additionally, we conduct a probabilistic forecasting study with significance tests and compare the predictive performance against literature benchmarks. The proposed day-ahead (month-ahead) model leads to a 13% (9%) reduction in out-of-sample continuous ranked probability score (CRPS) compared with the best performing benchmark model, mainly due to adequate modeling of the volatility and heavy tails.