In this thesis we study the Klein-Gordon equation on asymptotically anti-de Sitter (aAdS) spacetimes with boundary conditions implemented by pseudodifferential operators (PDOs) of order up to 2. Using techniques of microlocal analysis and b-calculus, we prove two propagation of singularities theorems, one for pseudodifferential operators of non-positive order and one for PDOs of order 0 < k <= 2, and we establish a well-posedness result in certain twisted Sobolev spaces. In particular, we

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... s the existence and uniqueness of the advanced and retarded fundamental solutions for the Klein-Gordon equation with prescribed boundary conditions and we characterize their wavefront set. At last, as a concrete application, we build the fundamental solutions for a massless Klein-Gordon equation on a static aAdS spacetime with admissible boundary conditions using the formalism of boundary triples.