Data-driven forecasts of air quality have recently achieved more accurate short-term predictions. Despite their success, most of the current data-driven solutions lack proper quantifications of model uncertainty that communicate how much to trust the forecasts. Recently, several practical tools to estimate uncertainty have been developed in probabilistic deep learning. However, there have not been empirical applications and extensive comparisons of these tools in the domain of air quality forecasts. Therefore, this work applies state-of-the-art techniques of uncertainty quantification in a real-world setting of air quality forecasts. Through extensive experiments, we describe training probabilistic models and evaluate their predictive uncertainties based on empirical performance, reliability of confidence estimate, and practical applicability. We also propose improving these models using "free" adversarial training and exploiting temporal and spatial correlation inherent in air quality data. Our experiments demonstrate that the proposed models perform better than previous works in quantifying uncertainty in data-driven air quality forecasts. Overall, Bayesian neural networks provide a more reliable uncertainty estimate but can be challenging to implement and scale. Other scalable methods, such as deep ensemble, Monte Carlo (MC) dropout, and stochastic weight averaging-Gaussian (SWAG), can perform well if applied correctly but with different tradeoffs and slight variations in performance metrics. Finally, our results show the practical impact of uncertainty estimation and demonstrate that, indeed, probabilistic models are more suitable for making informed decisions. Code and dataset are available at \url{https://github.com/Abdulmajid-Murad/deep_probabilistic_forecast}
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