There are other algebraic groups, but Chevalley's structure theorem asserts that every algebraic group is an extension of an abelian variety by a linear algebraic group. More precisely, if K is a perfect field, and G an algebraic group over K, there exists a unique normal closed subgroup H in G, such that H is a linear algebraic group and G/H an abelian variety.
According to another basic theorem[which?], any group that is also an affine variety has a faithful finite-dimensional linear representation: it is isomorphic to a matrix group, defined by polynomial equations.
Over the fields of real and complex numbers, every algebraic group is also a Lie group, but the converse is false
This may also be generalized by allowing schemes in place of varieties. The main effect of this in practice, apart from allowing subgroups in which the connected component is of finite index > 1, is to admit non-reduced schemes, in characteristic p.
There are a number of mathematical notions to study and classify algebraic groups.