  We study solutions of the Klein-Gordon, Maxwell, and linearized Einstein
equations in $\mathbb{R}^{1,d+1}$ that transform as $d$-dimensional conformal
primaries under the Lorentz group $SO(1,d+1)$. Such solutions, called conformal
primary wavefunctions, are labeled by a conformal dimension $\Delta$ and a
point in $\mathbb{R}^d$, rather than an on-shell $(d+2)$-dimensional momentum.
We show that the continuum of scalar conformal primary wavefunctions on the
principal continuous series $\Delta\in \frac d2+ i\mathbb{R}$ of $SO(1,d+1)$
spans a complete set of normalizable solutions to the wave equation. In the
massless case, with or without spin, the transition from momentum space to
conformal primary wavefunctions is implemented by a Mellin transform. As a
consequence of this construction, scattering amplitudes in this basis transform
covariantly under $SO(1,d+1)$ as $d$-dimensional conformal correlators.
