Is there a field of research based around music and emotion in the brain? I'm looking for research surrounding the scientific study of music and emotion and its effects on the brain. In particular, are there known parts of the brain which activate when playing or listening to music? 

Is there a specific field of study based around music and cognitive science?
Well it depends what you mean by "field of study", but yes there is significant Cognitive Science research on Music and emotional arousal and/or brain activity. In particular Cognitive Neuroscience is most relevant to actually mapping brain activations involved with various forms of music appreciation and creation.

A good study specifically on music and emotion is Investigating Emotion With Music:
An fMRI Study by Stefan Koelsch et all. They investigate listening to music and have nice heatmaps of brain activity during listening. There's many other papers exploring the activity pattern when listening or creating music, but the conclusion of this article sums up some of the major areas:


  In the present study, activations during the perception of
  unpleasant music were observed within an extensive neuronal network of limbic and paralimbic structures comprising the amygdala, hippocampus, parahippocampal gyrus,
  and temporal poles. Although clear increases of BOLD signals were observed in those structures in response to unpleasant music, strong signal decreases were also measured
  in response to pleasant music. This indicates that these
  structures respond to both pleasant and unpleasant auditory
  information with emotional valence, and that listening to
  music has the capacity to up- as well as down-regulate
  neuronal activity in these structures.
  
  ...
  
  During the presentation of the pleasant music, activations
  were observed in the ventral striatum, the anterior superior
  insula, and in the Rolandic operculum.


So interestingly, music with different emotional context results in different patterns and locations of brain activation. The full article has more specific information, and you can feel free to ask about more specific patterns of activation in regards to more specific stimuli.