Semantic Communication (SC) is an emerging technology aiming to surpass the Shannon limit. Traditional SC strategies often minimize signal distortion between the original and reconstructed data, neglecting perceptual quality, especially in low Signal-to-Noise Ratio (SNR) environments. To address this issue, we introduce a novel Generative AI Semantic Communication (GSC) system for single-user scenarios. This system leverages deep generative models to establish a new paradigm in SC. Specifically, At the transmitter end, it employs a joint source-channel coding mechanism based on the Swin Transformer for efficient semantic feature extraction and compression. At the receiver end, an advanced Diffusion Model (DM) reconstructs high-quality images from degraded signals, enhancing perceptual details. Additionally, we present a Multi-User Generative Semantic Communication (MU-GSC) system utilizing an asynchronous processing model. This model effectively manages multiple user requests and optimally utilizes system resources for parallel processing. Simulation results on public datasets demonstrate that our generative AI semantic communication systems achieve superior transmission efficiency and enhanced communication content quality across various channel conditions. Compared to CNN-based DeepJSCC, our methods improve the Peak Signal-to-Noise Ratio (PSNR) by 17.75% in Additive White Gaussian Noise (AWGN) channels and by 20.86% in Rayleigh channels.