We introduce the task of open-vocabulary visual instance search (OVIS). Given an arbitrary textual search query, Open-vocabulary Visual Instance Search (OVIS) aims to return a ranked list of visual instances, i.e., image patches, that satisfies the search intent from an image database. The term ``open vocabulary'' means that there are neither restrictions to the visual instance to be searched nor restrictions to the word that can be used to compose the textual search query. We propose to address such a search challenge via visual-semantic aligned representation learning (ViSA). ViSA leverages massive image-caption pairs as weak image-level (not instance-level) supervision to learn a rich cross-modal semantic space where the representations of visual instances (not images) and those of textual queries are aligned, thus allowing us to measure the similarities between any visual instance and an arbitrary textual query. To evaluate the performance of ViSA, we build two datasets named OVIS40 and OVIS1600 and also introduce a pipeline for error analysis. Through extensive experiments on the two datasets, we demonstrate ViSA's ability to search for visual instances in images not available during training given a wide range of textual queries including those composed of uncommon words. Experimental results show that ViSA achieves an mAP@50 of 27.8% on OVIS40 and achieves a recall@30 of 21.3% on OVIS1400 dataset under the most challenging settings.