Since strong convective weather is closely related to heavy precipitation, the nowcasting of convective weather, especially the nowcasting based on weather radar data, plays an essential role in meteorological operations for disaster prevention and mitigation. The traditional optical flow method and cross-correlation method have a low forecast accuracy and a short forecast leading time, while deep learning methods show remarkable advantages in nowcasting. However, most of the current forecasting methods based on deep learning suffer from the drawback that the forecast results become increasingly blurred as the forecast time increases. In this study, a weather radar nowcasting method based on the Temporal and Spatial Generative Adversarial Network (TSGAN) is proposed, which can obtain accurate forecast results, especially in terms of spatial details, by extracting spatial-temporal features, combining attention mechanisms and using a dual-scale generator and a multi-scale discriminator. The case studies on the forecast results of strong convective weather demonstrate that the GAN method performs well in terms of forecast accuracy and spatial detail representation compared with traditional optical flow methods and popular deep learning methods. Therefore, the GAN method proposed in this study can provide strong decision support for forecasting heavy precipitation processes. At present, the proposed method has been successfully applied to the actual weather forecasting business system.