Deep learning, a branch of artificial intelligence, has achieved unprecedented performance in several domains including medicine to assist with efficient diagnosis of diseases, prediction of disease progression and pre-screening step for physicians. Due to its significant breakthroughs, deep learning is now being used for the diagnosis of arthritis, which is a chronic disease affecting young to aged population. This paper provides a survey of recent and the most representative deep learning techniques (published between 2018 to 2020) for the diagnosis of osteoarthritis and rheumatoid arthritis. The paper also reviews traditional machine learning methods (published 2015 onward) and their application for the diagnosis of these diseases. The paper identifies open problems and research gaps. We believe that deep learning can assist general practitioners and consultants to predict the course of the disease, make treatment propositions and appraise their potential benefits.