With advanced imaging techniques, we are collecting images of various complex structures such as neurons, vessels, tissues and cells. These structures encode important information about underlying biological mechanisms. To fully exploit these structures, we propose to enhance learning pipelines with topology, the branch of abstract mathematics that deals with structures such as connections, loops and branches. Under-the-hood is a formulation of the topological computation as a robust and differentiable operator. This inspires a series of novel methods for segmentation, uncertainty estimation, generation, and analysis of these topology-rich biomedical structures. We demonstrate how these methods provide a better AI support for the annotation and analysis of images of vasculature and breast tissue. In digital pathology, we demonstrate how combining rich spatial and topological characterization with deep learning techniques will enhance cell/gland segmentation, synthesis, and gene expression prediction.