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Anaffine varietywithanactionof a semisimple groupGis called “small” if every nontrivial G-orbit in X is isomorphic to the orbit of a highest weight vector. Such a variety X carries a canonical action of the multiplicative group K∗ commuting with the G-action.We show that X is determined by the K∗-variety XU of fixed points under a maximal unipotent subgroup U ⊂ G. Moreover, if X is smooth, then X is a G-vector bundle over the algebraic quotient X//G. If G is of type An (n ≥ 2), Cn, E6, E7, or E8, we show that all affine G-varieties up to a certain dimension are small. As a consequence, we have the following result. If n ≥ 5, every smooth affine SLn-variety of dimension < 2n − 2 is an SLn-vector bundle over the smooth quotient X// SLn, with fiber isomorphic to the natural representation or its dual.