Shigeru Mukai is working on the moduli theory of algebraic varieties and vector bundles, and applying it to the study of algebraic varieties such as K3 surfaces, Enriques surfaces and Fano varieties. For example he classified 3-dimensional Fano varieties using vector bundles of higher rank and their deformation. In connection with moduli construction, he is interested in invariant theory. He poses the following problem in his study of Nagata's counter-example of Hilbert's fourteenth problem. "Are the invariant rings finitely generated when the 2-dimensional additive group acts linearly on polynomial rings?" Recently he is working on Enriques surfaces using their root systems governing the arrangements of smooth rational curves on them. Jointly with H. Ohashi, he recently generalized the relation between the Mathieu groups and finite symplectic automorphisms of K3 surfaces to Enriques surfaces.