Branching laws describe how irreducible representations of a group G decompose when restricted to a subgroup H. I will first give a short introduction to this topic in the context of real reductive groups and their unitary representations. Then I will explain how branching laws are intimately related to the spectral decomposition of certain differential operators, and why they are difficult to obtain in general. Finally I will present a method to obtain new branching laws from old ones using an analytic continuation procedure.