We consider the Helmholtz and the time-harmonic Maxwell transmission problems with one penetrable star-shaped Lipschitz obstacle. Under a natural assumption about the ratio of the wavenumbers, we prove bounds on the solution in terms of the data, with these bounds explicit in all parameters. For example, in the case of the Helmholtz equation, the (weighted) H1norm of the solution is bounded by the L2norm of the source term, independently of the wavenumber. Such explicit bounds are key to developing frequency-explicit error analysis for numerical methods such as FEM and BEM. The "shape-robustness" allows to quantify how variations in the shape of the obstacle affect the solution and makes the bounds particularly suitable for uncertainty quantification (UQ) applications. These bounds then imply the existence of a resonance-free strip beneath the real axis. If the assumption on the wavenumbers is lifted, then no bound with polynomial dependence on the wavenumber is possible. Joint work with Euan Spence (University of Bath).