A character variety of a finitely presented group 
and a complex reductive group 
is the affine quotient of its representation variety by conjugation.
Its coordinate ring is the ring of conjugation-invariant
functions on 
.
For 
,
its complex points correspond to semisimple representations up to isomorphism.
Representations with the same semisimplification therefore determine the same point.
For a complex algebraic curve 
, one takes 
. The 
-representation equations and conjugation invariants give
a natural integral model 
over 
.
The coprime fixed-determinant nonabelian Hodge correspondence uses a twisted character variety with product of commutators equal to a central scalar. Its mixed-Hodge weight filtration is compared with the perverse Leray filtration of the trace-free Hitchin fibration in the P=W conjecture.
A matrix is quasi-unipotent when every eigenvalue is a root of unity; the matrix need not be semisimple.
