Network latency prediction is important for server selection and quality-of-service estimation in real-time applications on the Internet. Traditional network latency prediction schemes attempt to estimate the latencies between all pairs of nodes in a network based on sampled round-trip times, through either Euclidean embedding or matrix factorization. However, these schemes become less effective in terms of estimating the latencies of personal devices, due to unstable and time-varying network conditions, triangle inequality violation and the unknown ranks of latency matrices. In this paper, we propose a matrix completion approach to network latency estimation. Specifically, we propose a new class of low-rank matrix completion algorithms, which predicts the missing entries in an extracted “network feature matrix” by iteratively minimizing a weighted Schatten- $p$ norm to approximate the rank. Simulations on true low-rank matrices show that our new algorithm achieves better and more robust performance than multiple state-of-the-art matrix completion algorithms in the presence of noise. We further enhance latency estimation based on multiple “frames” of latency matrices measured in the past, and extend the proposed matrix completion scheme to the case of 3-D tensor completion. Extensive performance evaluations driven by real-world latency measurements collected from the Seattle platform show that our proposed approaches significantly outperform various state-of-the-art network latency estimation techniques, especially for networks that contain personal devices.