Histopathological images contain rich phenotypic information that can be used to monitor underlying mechanisms contributing to disease progression and patient survival outcomes. Recently, deep learning has become the mainstream methodological choice for analyzing and interpreting histology images. In this paper, we present a comprehensive review of state-of-the-art deep learning approaches that have been used in the context of histopathological image analysis. From the survey of over 130 papers, we review the field's progress based on the methodological aspect of different machine learning strategies such as supervised, weakly supervised, unsupervised, transfer learning and various other sub-variants of these methods. We also provide an overview of deep learning based survival models that are applicable for disease-specific prognosis tasks. Finally, we summarize several existing open datasets and highlight critical challenges and limitations with current deep learning approaches, along with possible avenues for future research.