I will introduce some aspects of Conformal Field Theory (CFT) like energy-momentum tensor, conformal Ward identities, operator product expansions, and the Virasoro algebra by focusing on computations. Then I attempt to highlight where these fit into the mathematical approaches to CFT.
Links:
[1] http://drupal.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://drupal.mpim-bonn.mpg.de/node/4234
[3] http://drupal.mpim-bonn.mpg.de/node/12745