Image analysis is a central tool in modern biology. Cell and developmental biologists generate multidimensional microscopy data, including imaging of cellular, subcellular and tissue structures, in three dimensions, over time, and with multiple molecular markers. Segmentation and tracking of multidimensional microscopy data requires high accuracy across many images (e.g. timepoints) and is a labour-intensive part of biological image processing pipelines. We present ReSCU-Nets, recurrent convolutional neural networks that use the segmentation results from the previous frame as a prompt to segment the current frame. We demonstrate that ReSCU-Nets outperform state-of-the-art segmentation models in different tasks on biological multidimensional microscopy sequences.