Partial label learning (PLL) is an important problem that allows each training example to be labeled with a coarse candidate set with the ground-truth label included. However, in a more practical but challenging scenario, the annotator may miss the ground-truth and provide a wrong candidate set, which is known as the noisy PLL problem. To remedy this problem, we propose the PiCO+ framework that simultaneously disambiguates the candidate sets and mitigates label noise. Core to PiCO+, we develop a novel label disambiguation algorithm PiCO that consists of a contrastive learning module along with a novel class prototype-based disambiguation method. Theoretically, we show that these two components are mutually beneficial, and can be rigorously justified from an expectation-maximization (EM) algorithm perspective. To handle label noise, we extend PiCO to PiCO+, which further performs distance-based clean sample selection, and learns robust classifiers by a semi-supervised contrastive learning algorithm. Beyond this, we further investigate the robustness of PiCO+ in the context of out-of-distribution noise and incorporate a novel energy-based rejection method for improved robustness. 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.