It is now generally recognized that user-provided image tags are incomplete and noisy. In this study, we focus on the problem of tag completion that aims to simultaneously enrich the missing tags and remove noisy tags. The novel component of the proposed framework is a noisy matrix recovery algorithm. It assumes that the observed tags are independently sampled from an unknown tag matrix and our goal is to recover the tag matrix based on the sampled tags. We show theoretically that the proposed noisy tag matrix recovery algorithm is able to simultaneously recover the missing tags and de-emphasize the noisy tags even with a limited number of observations. In addition, a graph Laplacian based component is introduced to combine the noisy matrix recovery component with visual features. Our empirical study with multiple benchmark datasets for image tagging shows that the proposed algorithm outperforms state-of-the-art approaches in terms of both effectiveness and efficiency when handling missing and noisy tags.