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Partial label learning (PLL) is an important problem that allows each training example to be labeled with a coarse candidate set, which well suits many real-world data annotation scenarios with label ambiguity. Despite the promise, the performance of PLL often lags behind the supervised counterpart. In this work, we bridge the gap by addressing two key research challenges in PLL -- representation learning and label disambiguation -- in one coherent framework. Specifically, our proposed framework PiCO consists of a contrastive learning module along with a novel class prototype-based label disambiguation algorithm. PiCO produces closely aligned representations for examples from the same classes and facilitates label disambiguation. Theoretically, we show that these two components are mutually beneficial, and can be rigorously justified from an expectation-maximization (EM) algorithm perspective. Moreover, we study a challenging yet practical noisy partial label learning setup, where the ground-truth may not be included in the candidate set. To remedy this problem, we present an extension PiCO+ that performs distance-based clean sample selection and learns robust classifiers by a semi-supervised contrastive learning algorithm. Extensive experiments demonstrate that our proposed methods significantly outperform the current state-of-the-art approaches in standard and noisy PLL tasks and even achieve comparable results to fully supervised learning.