Summary: Metaharmonic wavelets are introduced for constructing the solution of the Helmholtz equation (reduced wave equation) corresponding to Dirichlet’s or Neumann’s boundary values on a closed surface \(\Sigma\) in three-dimensional Euclidean space \(\mathbb{R}^3\). A consistent scale continuous and scale discrete wavelet approach leading to exact reconstruction formulas is considered in more detail. A scale discrete version of multiresolution is described for potential functions metaharmonic outside the closed surface and satisfying the radiation condition at infinity. Moreover, we discuss fully discrete wavelet representations of band-limited metaharmonic potentials. Finally, a decomposition and reconstruction (pyramid) scheme for economical numerical implementation is presented for Runge-wavelet approximation.