Yes, there is research on this topic. The field that studies it is known as affective computing (AC). Emotion recognition seems to be a specific problem in affective computing, i.e. the recognition of emotions, while AC is also concerned with giving machines the ability to convey emotions (in fact, this paper differentiates the two). There's also sentiment analysis, which refers to the analysis of the sentiment of people e.g. in social media (e.g. in comments) using natural language processing techniques, which doesn't seem exactly what you are looking for. These fields/tasks are all very related/similar, but I don't think they are exactly synonymous (although I'm not an expert in this topic).
Apart from the linked Wikipedia articles, you can find another list of related resources (including important papers and books, and software) here. If you're interested in an easy overview of the AC field, you should probably read the paper Affective computing: challenges, by Rosalind W. Picard (who is one of the leading researchers in this field). In this paper, she differentiates between emotions (what you observe/express with gestures, facial expressions, etc.) and feelings (internal states, which sometimes are not even clear to the person that is having them), and compares emotions to the weather: more precisely, when you say that the weather is windy, you're just giving a label to a fuzzy set of conditions: for example, there's the wind, it could start raining soon, etc., but these conditions are not always clear: for instance, how strong should the wind be in order for you to consider the weather windy? I found this is a nice analogy that conveys what emotions are and why interpreting emotions seems to be a difficult task.
Additionally, you may also be interested in the loving AI project.