Rationality of algebraic groups The Cayley parametrization of $O(n),$ as in my answer to this question makes one wonder: which algebraic groups are actually rational? I am sure this is very well understood, just not by me...
 A: A reductive group $G$ over a ground field $k$ need not be a rational variety over $k$ (although the group $G_{/\ell}$ obtained by scalar extension is a rational $\ell$ variety, where $\ell$ is an algebraic closure of $k$). Maybe a good place to look concerning these matters is Merkurjev's 1996 Publ. Math. IHES paper "R-equivalence and rationality problem for semisimple adjoint classical algebraic groups".
In the intro to that paper, Merkurjev makes some interesting observations:


*

*Chevalley showed in the 50s that there are algebraic tori over local fields which are not rational varieties.

*algebraic tori over dimension <= 2 are always rational varieties; thus, any reductive
group of rank <=2 is rational. (Chevalley proved that the variety of maximal tori in a connected group is a rational variety, so the rationality of $G$ boils down to the rationality of some maximal torus of $G$).

*any semisimple group over a number field which is a counterexample to weak approximation is an example of a non-rational group.


In Merkurjev's paper, you will find explicit examples of groups $G$ which are not stably rational -- a $k$-variety $X$ is stably rational if $X \times \operatorname{Aff}^d$ is a rational $k$-variety for some $d \ge 0$. 
One way to show that $G$ is not stably rational (and hence not rational) is to show that
the "group of $R$-equivalence classes" for $G$ is non-trivial;
this group of $R$-equivalence classes was introduced by Manin. Merkurjev's paper is devoted to the computation of the group of $R$-equivalence classes for semisimple, adjoint, classical groups.
A: The following answer applies to affine algebraic groups over algebraically closed fields, or more generally for quasi-split groups, as Scott Carnahan suggests.  I don't really know about these rationality (in the number-theoretic sense) questions.
Under these conditions, every algebraic group is an extension of a reductive group by a unipotent group (since a reductive group is one whose unipotent radical, i.e. maximal normal unipotent connected subgroup, is trivial).  Every unipotent group is rational, since it is isomorphic as a variety to $\mathbb{A}^n$.  Every reductive group is rational by the Bruhat decomposition.  Every Zariski-locally trivial bundle over a rational variety whose fibers are rational, is rational.
A: Suppose G is a connected affine algebraic group over a field k. If G is reductive or k is perfect, then G is unirational. A reference is Borel, Linear Algebraic Groups, Theorem 18.2. This doesn't quite give rationality, but is the best result in general that I know of, and is sufficient for some applications. (Ryan's answer of course gives more precise information under stronger hypotheses).
An affine counterexample over an imperfect field of characteristic p is given by y^p=x+tx^p where t is not a p-th power in k. (This is a subgroup of Ga2).
Now if G is connected and not affine, then it surjects onto an abelian variety. So if G were rational, then we could get a nonconstant morphism from P^1 to an abelian variety. Now let me just quote https://math.stackexchange.com/a/114611 so no non-affine conntected G can be rational, or even unirational.
