We describe manifestly globally supersymmetric theories in five dimensions and their dimensional reduction in superspace. This proceeds by a reduction from harmonic superspace, the universal superspace for theories in dimension six or fewer with eight supercharges, through projective superspace to simple superspace. That is, all symmetries of the original theory are retained but only a proper subgroup of the original Lorentz supergroup is realized linearly. The latter supergroup is precisely the four-dimensional Lorentz supergroup with four supercharges, making this setting ideal for the study of particle-field theories in five or six dimensions with flat four-dimensional subspaces.