Questions about algebraic structures known as quantum groups, and their categories of representations. Quasitriangular Hopf algebras and their Drinfel'd twists, triangular Hopf algebras, $C^\star$ quantum groups, h-adic quantum groups, various semisimplified categories at roots of unity which are called "quantum groups", bicrossproduct quantum groups, and quantum groups coming from braided tensor categories.
Quantum groups describe various different but broadly related types of noncommutative algebras with additional structure. The first quantum groups came from the theory of quantum integral systems, and were formalized by Drinfel'd and Jimbo as noncommutative deformations of universal enveloping algebras of semisimple Lie algebras, or more generally of Kac-Moody Lie algebras. The h-adic completion of the resulting algebra has the structure of a quasitriangular Hopf algebra. Infinite dimensional generalizations include $C^\star$ quantum groups (Woronowicz) and h-adic quantum groups. Another class of quantum groups are Majid's bicrossproduct quantum groups, which arise as deformations of solvable Lie groups.
Categories of representations of quantum groups admit braided tensor structures, which make them useful for constructing low dimensional topological invariants. In this vein, there are several related semisimplified categories at roots of unity which occur in Reshetikhin-Turaev constructions of quantum invariants, and which are also called quantum groups.
Further information on quantum groups, their intuitions, and their motivations, may be found in Majid's AMS Notices survey article.