Schema matching is a crucial task in data integration, involving the alignment of a source database schema with a target schema to establish correspondence between their elements. This task is challenging due to textual and semantic heterogeneity, as well as differences in schema sizes. Although machine-learning-based solutions have been explored in numerous studies, they often suffer from low accuracy, require manual mapping of the schemas for model training, or need access to source schema data which might be unavailable due to privacy concerns. In this paper we present a novel method, named ReMatch, for matching schemas using retrieval-enhanced Large Language Models (LLMs). Our method avoids the need for predefined mapping, any model training, or access to data in the source database. In the ReMatch method the tables of the target schema and the attributes of the source schema are first represented as structured passage-based documents. For each source attribute document, we retrieve $J$ documents, representing target schema tables, according to their semantic relevance. Subsequently, we create a prompt for every source table, comprising all its attributes and their descriptions, alongside all attributes from the set of top $J$ target tables retrieved previously. We employ LLMs using this prompt for the matching task, yielding a ranked list of $K$ potential matches for each source attribute. Our experimental results on large real-world schemas demonstrate that ReMatch significantly improves matching capabilities and outperforms other machine learning approaches. By eliminating the requirement for training data, ReMatch becomes a viable solution for real-world scenarios.