Examination of tissue biopsy and quantification of the various characteristics of cellular processes are clinical benchmarks in cancer diagnosis. Nuclei and glands instance segmentation greatly assists the high-throughput quantification of cellular process and accurate appraisal of tissue biopsy. It subsequently makes a significant improvement to the computational pathology process for cancer diagnosis, treatment planning, and survival analysis. Recent advancements in the field of computer vision have automated the manual, laborious, and time-consuming histopathological analysis process. Automated image analysis of histopathological images for cells and tissues to trace the entirety of the ultrastructures, has been an active area of research in medical informatics for decades. The developments in computers, microscopy hardware, and the availability of large-scale public datasets have further fastened the development in this field. And the realization that scientific and diagnostic pathology calls for fresh ways to undertake, automated image analysis of histopathological images has captivated contemporary attention. In this survey, 126 papers illustrating the AI-based methods for nuclei and glands instance segmentation published in the last five years (2017–2022) are deeply analyzed, and the limitations of current approaches and the open challenges are discussed. Moreover, the potential future research direction is presented, and the contribution of state-of-the-art methods is summarized. Further, a generalized summary of publicly available datasets and detailed insights on the grand challenges illustrating the top-performing methods specific to each challenge is also provided. Besides, we intended to give the reader the current state of existing research and pointers to the future directions in developing methods that can be used in clinical practice enabling improved diagnosis, grading, prognosis, and treatment planning of cancer. To the best of our knowledge, no previous work has reviewed the instance segmentation in histology images focusing on nuclei and glands instance segmentation.