In the era of big data, text-based medical data, such as electronic health records (EHR) and electronic medical records (EMR), are growing rapidly. EHR and EMR are collected from patients to record their basic information, lab tests, vital signs, clinical notes, and reports. EHR and EMR contain the helpful information to assist oncologists in computer-aided diagnosis and decision making. However, it is time consuming for doctors to extract the valuable information they need and analyze the information from the EHR and EMR data. Recently, more and more research works have applied natural language processing (NLP) techniques, i.e., rule-based, machine learning-based, and deep learning-based techniques, on the EHR and EMR data for computer-aided diagnosis in oncology. The objective of this review is to narratively review the recent progress in the area of NLP applications for computer-aided diagnosis in oncology. Moreover, we intend to reduce the research gap between artificial intelligence (AI) experts and clinical specialists to design better NLP applications. We originally identified 295 articles from the three electronic databases: PubMed, Google Scholar, and ACL Anthology; then, we removed the duplicated papers and manually screened the irrelevant papers based on the content of the abstract; finally, we included a total of 23 articles after the screening process of the literature review. Furthermore, we provided an in-depth analysis and categorized these studies into seven cancer types: breast cancer, lung cancer, liver cancer, prostate cancer, pancreatic cancer, colorectal cancer, and brain tumors. Additionally, we identified the current limitations of NLP applications on supporting the clinical practices and we suggest some promising future research directions in this paper.