Analysis of stress, depression and engagement is less common and more complex than that of frequently discussed emotions such as happiness, sadness, fear and anger. The importance of these psychological states has been increasingly recognized due to their implications for mental health and well-being. Stress and depression are interrelated and together they impact engagement in daily tasks, highlighting the need to explore their interplay. This survey is the first to simultaneously explore computational methods for analyzing stress, depression and engagement. We present a taxonomy and timeline of the computational approaches used to analyze them and we discuss the most commonly used datasets and input modalities, along with the categories and generic pipeline of these approaches. Subsequently, we describe state-of-the-art computational approaches, including a performance summary on the most commonly used datasets. Following this, we explore the applications of stress, depression and engagement analysis, along with the associated challenges, limitations and future research directions.