We examine possible vacuum states for scalar fields in de Sitter space, concentrating on those states (1) invariant under the de Sitter group O(1,4) or (2) invariant under one of its maximal subgroups E(3). For massive fields there is a one-complex-parameter family of de Sitter-invariant states, which includes the ``Euclidean'' vacuum state as a special case. We show these states are generated from the Euclidean vacuum by a frequency-independent Bogoliubov transformation, and obtain formulas for the symmetric, antisymmetric, and Feynman functions. In the massless minimally coupled case we prove that there exists no de Sitter-invariant Fock vacuum state. However one can find Fock states which are E(3) invariant. These states include the Bunch-Davies and Ottewill-Najmi vacua as special cases.