Contrastive Language-Image Pre-training (CLIP) models as a milestone of modern multimodal intelligence, its generalization mechanism grasped massive research interests in the community. While existing studies limited in the scope of pre-training knowledge, hardly underpinned its generalization to countless open-world concepts absent from the pre-training regime. This paper dives into such Out-of-Pre-training (OOP) generalization problem from a holistic perspective. We propose LAION-Beyond benchmark to isolate the evaluation of OOP concepts from pre-training knowledge, with regards to OpenCLIP and its reproducible variants derived from LAION datasets. Empirical analysis evidences that despite image features of OOP concepts born with significant category margins, their zero-shot transfer significantly fails due to the poor image-text alignment. To this, we elaborate the name-tuning'' methodology with its theoretical merits in terms of OOP generalization, then propose few-shot name learning (FSNL) and zero-shot name learning (ZSNL) algorithms to achieve OOP generalization in a data-efficient manner. Their superiority have been further verified in our comprehensive experiments.