I am wondering what currently our mathematicians do related to conformal field theory, (I know currently it is a central topic, but I have only a vague idea what mathematicians do in there), or more generally topological field theory, field theory... I am extremely appreicated if there is any survey paper.
CFT/QFT/TFT/etc. is a huge subject... Here are some random references off the top of my head... Segal, "The definition of conformal field theory". Costello, "Topological conformal field theories and CalabiYau categories"  This is (essentially) the 2d version of the (Hopkins)Lurie/BaezDolan cobordism hypothesis that Lennart mentions. See also KontsevichSoibelman, "Notes on Ainfinity...". This stuff is closely related to mirror symmetry, which is  in physics terms  a duality between certain field theories (or sigma models). Mirror symmetry by itself is already a huge enterprise... See papers by YiZhi Huang for stuff about vertex operator algebras and CFTs. One can consider string topology from a field theory viewpoint... see for example Sullivan, "String Topology: Background and Present State" and BlumbergCohenTeleman, "Openclosed field theories, string topology, and Hochschild homology". This is actually related to the work of Costello, Lurie, Kontsevich mentioned above  see e.g. section 2.1 of Costello's paper. An important problem is that of making rigorous some of the things that physicists do in QFT, such as path integrals. See Costello, "Renormalization and effective field theory" and also Borcherds, "Renormalization and quantum field theory". There's also ChernSimons theory... GromovWitten theory... KapustinWitten theory... RozanskyWitten theory... Related MO questions: A reading list for topological quantum field theory? 


It is a huge topic (or a more than one huge topic) and I am not an expert, but I want to offer two examples of what is currently en vogue in topological field theories: 1) Extended Field Theories: Classicly you give for every ndimensional a vector space and for every (n+1)dimensional bordism a morphism. Extended can mean that you replace the category of vector spaces by a suitable 2category and give for every bordism between bordisms a 2morphism. For more precise statements and a fancy classification result, you should have a look at: http://www.math.harvard.edu/~lurie/papers/cobordism.pdf 2) Another important topic is the connection between 2dimensional conformal field theories and 3dimension topological field theories  or more generally: how does one construct 3dimensional topological field theories? A good introduction is the book of Bakalov and Kirillov  Lectures on Tensor Categories and Modular Functors. 


Note that the word "conformal field theory" (indeed, "field theory" in general) has many meanings, depending on the area of mathematics of the user, and in general the relationships between the different meanings is only conjectural. For one use of the word, with applications to the representation theory of loop groups, you might check out the notes form the recent Workshop on Operator Algebras and Conformal Field Theory at University of Oregon. 


If you are interested in operator algebras, try this review by Kawahigashi From Operator Algebras to Superconformal Field Theory, where connections to subfactor theory, moonshine and noncommutative geometry are pointed out. 


One direction is the attempt of Igor Kriz to make conformal field theory rigorous in terms of lax algebras and elliptic cohomology. This work also addresses the interesting problem of understanding the moduli space of rational conformal field theories, a problem that's also important in string theory, where certain types of rational conformal field theories can be related to CalabiYau varieties. The deformations of the CFT then are to be mapped to the deformations of the CalabiYau families. This raises interesting problems because of the singularities encountered in the CY moduli space. Papers in this direction can be found on his webpage at Michigan. More generally, there is much work done by Bloch, Marcolli and others to understand the appearance of motives in perturbative quantum field theory. Papers in this direction can be found on Bloch's Chicago webpage and Marcolli's Caltech webpage. Marcolli has also recently written a book on this theme called "Feynman Motives". 

