Imposing key anatomical features, such as the number of organs, their shapes and relative positions, is crucial for building a robust multi-organ segmentation model. Current attempts to incorporate anatomical features include broadening the effective receptive field (ERF) size with data-intensive modules, or introducing anatomical constraints that scales poorly to multi-organ segmentation. We introduce a novel architecture called the Anatomy-Informed Cascaded Segmentation Network (AIC-Net). AIC-Net incorporates a learnable input termed "Anatomical Prior", which can be adapted to patient-specific anatomy using a differentiable spatial deformation. The deformed prior later guides decoder layers towards more anatomy-informed predictions. We repeat this process at a local patch level to enhance the representation of intricate objects, resulting in a cascaded network structure. AIC-Net is a general method that enhances any existing segmentation models to be more anatomy-aware. We have validated the performance of AIC-Net, with various backbones, on three multi-organ segmentation tasks: abdominal organs, vertebrae, and ribs. For each respective task, our benchmarks demonstrate improved dice score and Hausdorff distance.