The quantitative analysis of digital pathology images plays a significant role in the diagnosis of benign and malignant diseases such as breast cancer and prostate cancer, in which the morphology measurements of histologic primitives serve as a basis of quantitative analyses. However, the complex nature of digital pathology data, such as diversity, present significant challenges for such segmentation task, which might lead to difficulties in feature extraction and instance segmentation. By converting complex pathology data into minable image features using artificial intelligence assisted pathologist's analysis, it becomes possible to automatically extract quantitative information of individual primitives. Machine learning algorithms, in particular deep models, are emerging as leading tools in quantitative analyses of digital pathology. It has exhibited great power in feature learning with producing improved accuracy of various tasks. This survey provides a comprehensive review of this fast-growing field. Popular deep models are briefly introduced, including convolutional neural networks, fully convolutional networks, encoder-decoder architectures, recurrent neural networks, and generative adversarial networks, and current deep learning achievements in various tasks are summarized, such as detection and segmentation. This study also presents the mathematical theory, key steps, main advantages and disadvantages, and performance analysis of deep learning algorithms, and interprets their formulations or modelings for specific tasks. In addition, the open challenges and potential trends of future research are discussed in pathology image segmentation using deep learning.