Cytology is a branch of pathology that deals with the microscopic examination of cells for diagnosis of carcinoma or inflammatory conditions. In the present work, the term cytology is used to indicate solid organ cytology. Automation in cytology started in the early 1950s with an aim to reduce manual efforts in the diagnosis of cancer. The influx of intelligent systems with high computational power and improved specimen collection techniques helped to achieve technological heights in the cytology automation process. In the present survey, we focus on image analysis techniques paving the way to automation in cytology. We take a short tour of 17 types of solid organ cytology to explore various segmentation and/or classification techniques that evolved during the past three decades to automate cytology image analysis. It is observed that most of the works are aligned toward three types of cytology: Cervical, Breast, and Respiratory tract cytology. These are discussed elaborately in the article. Commercial systems developed during the period are also summarized to comprehend the overall growth in respective domains. Finally, we discuss different state-of-the-art methods and related challenges to provide prolific and competent future research directions in bringing cytology-based commercial systems into the mainstream.