Large Generative Models (LGMs) such as GPT, Stable Diffusion, Sora, and Suno are trained on a huge amount of texts, images, videos, and audio that are extremely diverse from numerous domains. This large-scale training paradigm on diverse well-curated data enhances the creativity and diversity of the generated content. However, all previous graph-generative models (e.g., GraphRNN, MDVAE, MoFlow, GDSS, and DiGress) have been trained only on one dataset each time, which cannot replicate the revolutionary success achieved by LGMs in other fields. To remedy this crucial gap, we propose a large-scale training paradigm that uses a large corpus of graphs (over 5000 graphs) from 13 domains, leading to the development of large graph generative models (LGGMs). We empirically demonstrate that the pre-trained LGGMs have superior zero-shot generative capability to existing graph generative models. Furthermore, our pre-trained LGGMs can be easily fine-tuned with graphs from target domains and demonstrate even better performance than those directly trained from scratch, behaving as a solid starting point for real-world customization. Inspired by Stable Diffusion, we further equip LGGMs with the Text-to-Graph generation capability, such as providing the description of the network name and domain (i.e., "The power-1138-bus graph represents a network of buses in a power distribution system.") and network statistics (i.e., "The graph has a low average degree, suitable for modeling social media interactions."). This Text-to-Graph capability integrates the extensive world knowledge in the underlying language model, offering users fine-grained control of the generated graphs. We release the code, the model checkpoint, and the datasets at https://github.com/KINDLab-Fly/LGGM.