Let X be a complete, simply connected harmonic manifold of purely exponential volume growth. This class contains all non-flat harmonic manifolds of non-positive curvature and, in particular all known examples of non-compact harmonic manifolds except for the flat spaces. Denote by $$h > 0$$ the mean curvature of horospheres in X, and set $$\rho = h/2$$ . Fixing a basepoint $$o \in X$$ , for $$\xi \in \partial X$$ , denote by $$B_{\xi }$$ the Busemann function at $$\xi $$ such that $$B_{\xi }(o) = 0$$ . Then for $$\lambda \in \mathbb {C}$$ the function $$e^{(i\lambda - \rho )B_{\xi }}$$ is an eigenfunction of the Laplace–Beltrami operator with eigenvalue $$-(\lambda ^2 + \rho ^2)$$ . For a function f on X, we define the Fourier transform of f by $$\begin{aligned} \tilde{f}(\lambda , \xi ) := \int _X f(x) e^{(-i\lambda -\rho )B_{\xi }(x)} \mathrm{{d}}vol(x) \end{aligned}$$ for all $$\lambda \in \mathbb {C}, \xi \in \partial X$$ for which the integral converges. We prove a Fourier inversion formula $$\begin{aligned} f(x) = C_0 \int _{0}^{\infty } \int _{\partial X} \tilde{f}(\lambda , \xi ) e^{(i\lambda - \rho )B_{\xi }(x)} \mathrm{{d}}\lambda _o(\xi ) |c(\lambda )|^{-2} \mathrm{{d}}\lambda \end{aligned}$$ for $$f \in C^{\infty }_c(X)$$ , where c is a certain function on $$\mathbb {R} - \{0\}$$ , $$\lambda _o$$ is the visibility measure on $$\partial X$$ with respect to the basepoint $$o \in X$$ and $$C_0 > 0$$ is a constant. We also prove a Plancherel theorem, and a version of the Kunze–Stein phenomenon.