Abstract:  There are many problems in number theory, differential geometry, algebraic geometry, representation theory, and finite group theory where the notion of "reductive linear algebraic group over a field" naturally arises and provides insight as well as technique. The initial definition gives little indication that there should be an interesting theory! I'll explain some of the basic definitions and examples, and present several natural ways in which such groups arise and illuminate problems that don't seem on the surface to be about linear algebraic groups.

The assumed background is awareness of some definitions:  the Lie algebra of a Lie group (not of its nontrivial properties), and an affine algebraic variety over an algebraically closed field (such as the complex numbers, if you like that).