Let α be a Z^d-action (d≧ 2) by automorphisms of a compact metric abelian group. For any non-linear shape I⊂Z^d, there is an α with the property that I is a minimal mixing shape for α. The only implications of the form "I is a mixing shape for α ⇒ J is a mixing shape for α'' are trivial ones for which I contains a translate of J.

If all shapes are mixing for α, then α is mixing of all orders. In contrast to the algebraic case, if β is a Z^d-action by measure-preserving transformations, then all shapes mixing for β does not preclude rigidity.

Finally, we show that mixing of all orders in cones -- a property that coincides with mixing of all orders for Z-actions -- holds for algebraic mixing Z²-actions.

The author gratefully acknowledges support from NSF grant DMS-94-01093.