Face anti-spoofing is crucial to the security of face recognition systems. Most previous methods formulate face anti-spoofing as a supervised learning problem to detect various predefined presentation attacks, which need large scale training data to cover as many attacks as possible. However, the trained model is easy to overfit several common attacks and is still vulnerable to unseen attacks. To overcome this challenge, the detector should: 1) learn discriminative features that can generalize to unseen spoofing types from predefined presentation attacks; 2) quickly adapt to new spoofing types by learning from both the predefined attacks and a few examples of the new spoofing types. Therefore, we define face anti-spoofing as a zero- and few-shot learning problem. In this paper, we propose a novel Adaptive Inner-update Meta Face Anti-Spoofing (AIM-FAS) method to tackle this problem through meta-learning. Specifically, AIM-FAS trains a meta-learner focusing on the task of detecting unseen spoofing types by learning from predefined living and spoofing faces and a few examples of new attacks. To assess the proposed approach, we propose several benchmarks for zero- and few-shot FAS. Experiments show its superior performances on the presented benchmarks to existing methods in existing zero-shot FAS protocols.