Depth completion (DC) aims to predict a dense depth map from an RGB image and a sparse depth map. Existing DC methods generalize poorly to new datasets or unseen sparse depth patterns, limiting their real-world applications. We propose OMNI-DC, a highly robust DC model that generalizes well zero-shot to various datasets. The key design is a novel Multi-Resolution Depth Integrator, allowing our model to deal with very sparse depth inputs. We also introduce a novel Laplacian loss to model the ambiguity in the training process. Moreover, we train OMNI-DC on a mixture of high-quality datasets with a scale normalization technique and synthetic depth patterns. Extensive experiments on 7 datasets show consistent improvements over baselines, reducing errors by as much as 43%. Codes and checkpoints will be made public.