Few-shot learning (FSL) focuses on distilling transferrable knowledge from existing experience to cope with novel concepts for which the labeled data are scarce. A typical assumption in FSL is that the training examples of novel classes are all clean with no outlier interference. In many realistic applications where examples are provided by users, however, data are potentially noisy or unreadable. In this context, we introduce a novel research topic, robust FSL (RFSL), where we aim to address two types of outliers within user-provided data: the representation outlier (RO) and the label outlier (LO). Moreover, we introduce a metric for estimating robustness and use it to investigate the performance of several advanced methods to FSL when faced with user-provided outliers. In addition, we propose robust attentive profile networks (RapNets) to achieve outlier suppression. The results of a comprehensive evaluation of benchmark data sets demonstrate the shortcomings of current FSL methods and the superiority of the proposed RapNets when dealing with RFSL problems, establishing a benchmark for follow-up studies.