Creating diverse and high-quality 3D assets with an automatic generative model is highly desirable. Despite extensive efforts on 3D generation, most existing works focus on the generation of a single category or a few categories. In this paper, we introduce a diffusion-based feed-forward framework for synthesizing massive categories of real-world 3D objects \textit{with a single generative model}. Notably, there are three major challenges for this large-vocabulary 3D generation: \textbf{a}) the need for expressive yet efficient 3D representation; \textbf{b}) large diversity in geometry and texture across categories; \textbf{c}) complexity in the appearances of real-world objects. To this end, we propose a novel triplane-based 3D-aware \textbf{Diff}usion model with \textbf{T}rans\textbf{F}ormer, \textbf{DiffTF}, for handling challenges via three aspects. \textbf{1}) Considering efficiency and robustness, we adopt a revised triplane representation and improve the fitting speed and accuracy. \textbf{2}) To handle the drastic variations in geometry and texture, we regard the features of all 3D objects as a combination of generalized 3D knowledge and specialized 3D features. To extract generalized 3D knowledge from diverse categories, we propose a novel 3D-aware transformer with shared cross-plane attention. It learns the cross-plane relations across different planes and aggregates the generalized 3D knowledge with specialized 3D features. \textbf{3}) In addition, we devise the 3D-aware encoder/decoder to enhance the generalized 3D knowledge in the encoded triplanes for handling categories with complex appearances. Extensive experiments on ShapeNet and OmniObject3D (over 200 diverse real-world categories) convincingly demonstrate that a single DiffTF model achieves state-of-the-art large-vocabulary 3D object generation performance with large diversity, rich semantics, and high quality. More results are available at https://difftf.github.io/