A common practice in metric learning is to train and test an embedding model for each dataset. This dataset-specific approach fails to simulate real-world scenarios that involve multiple heterogeneous distributions of data. In this regard, we introduce a novel metric learning paradigm, called Universal Metric Learning (UML), which learns a unified distance metric capable of capturing relations across multiple data distributions. UML presents new challenges, such as imbalanced data distribution and bias towards dominant distributions. To address these challenges, we propose Parameter-efficient Universal Metric leArning (PUMA), which consists of a pre-trained frozen model and two additional modules, stochastic adapter and prompt pool. These modules enable to capture dataset-specific knowledge while avoiding bias towards dominant distributions. Additionally, we compile a new universal metric learning benchmark with a total of 8 different datasets. PUMA outperformed the state-of-the-art dataset-specific models while using about 69 times fewer trainable parameters.