The existence of noisy data is prevalent in both the training and testing phases of machine learning systems, which inevitably leads to the degradation of model performance. There have been plenty of works concentrated on learning with in-distribution (IND) noisy labels in the last decade, i.e., some training samples are assigned incorrect labels that do not correspond to their true classes. Nonetheless, in real application scenarios, it is necessary to consider the influence of out-of-distribution (OOD) samples, i.e., samples that do not belong to any known classes, which has not been sufficiently explored yet. To remedy this, we study a new problem setup, namely Learning with Open-world Noisy Data (LOND). The goal of LOND is to simultaneously learn a classifier and an OOD detector from datasets with mixed IND and OOD noise. In this paper, we propose a new graph-based framework, namely Noisy Graph Cleaning (NGC), which collects clean samples by leveraging geometric structure of data and model predictive confidence. Without any additional training effort, NGC can detect and reject the OOD samples based on the learned class prototypes directly in testing phase. We conduct experiments on multiple benchmarks with different types of noise and the results demonstrate the superior performance of our method against state of the arts.