Histopathology refers to the examination by a pathologist of biopsy samples. Histopathology images are captured by a microscope to locate, examine, and classify many diseases, such as different cancer types. They provide a detailed view of different types of diseases and their tissue status. These images are an essential resource with which to define biological compositions or analyze cell and tissue structures. This imaging modality is very important for diagnostic applications. The analysis of histopathology images is a prolific and relevant research area supporting disease diagnosis. In this paper, the challenges of histopathology image analysis are evaluated. An extensive review of conventional and deep learning techniques which have been applied in histological image analyses is presented. This review summarizes many current datasets and highlights important challenges and constraints with recent deep learning techniques, alongside possible future research avenues. Despite the progress made in this research area so far, it is still a significant area of open research because of the variety of imaging techniques and disease-specific characteristics.