Deep convolutional neural networks have driven substantial advancements in the automatic understanding of images. Requiring a large collection of images and their associated annotations is one of the main bottlenecks limiting the adoption of deep networks. In the task of medical image segmentation, requiring pixel-level semantic annotations performed by human experts exacerbate this difficulty. This paper proposes a new framework to train a fully convolutional segmentation network from a large set of cheap unreliable annotations and a small set of expert-level clean annotations. We propose a spatially adaptive reweighting approach to treat clean and noisy pixel-level annotations commensurately in the loss function. We deploy a meta-learning approach to assign higher importance to pixels whose loss gradient direction is closer to those of clean data. Our experiments on training the network using segmentation ground truth corrupted with different levels of annotation noise show how spatial reweighting improves the robustness of deep networks to noisy annotations.