The workshop “Geometric Aspects of Spectral Theory” brought together leading researchers working in various areas of this vast field of mathematics. The meeting featured presentations on some of the most fascinating recent developments in the subject, including five survey talks given by top experts, as well as reports on the progress made by graduate students and postdocs. A number of new stimulating questions were formulated during the open problem session. Mathematics Subject Classification (2000): 35J05, 35J10, 35J20, 58J05, 58J32, 58J50, 35P15, 35P20, 35P99. Introduction by the Organisers Geometric spectral theory is a rapidly developing area of mathematics with connections to Riemannian geometry, mathematical physics, calculus of variations and other fields. The talks presented at the meeting covered a broad variety of topics, already well-established as well as relatively unexplored. The workshop featured daily survey talks given by Brian Davies, Bernard Helffer, Bruno Colbois, Aldo Pratelli and Rupert Frank. On Wednesday evening, Timo Weidl gave a lecture aimed at a general mathematical audience, attended by the participants of both workshops that were held at the Institute during the week. Main topics of the meeting included optimisation problems for eigenvalues (talks by Aldo Pratelli, Dorin Bucur, Pedro Freitas, Mette Iversen, Richard Laugesen, and a related talk by Michael Loss), spectral properties of Dirac and Schrodinger operators (talks by Rupert Frank, Anna Dall’Acqua, Ari Laptev, Michael Levitin 2014 Oberwolfach Report 33/2012 and Timo Weidl), geometric estimates for Laplace and Steklov eigenvalues on Riemannian manifolds (talks by Bruno Colbois, Alexandre Girouard and Alessandro Savo), geometric features of nodal domains and spectral minimal partitions (talks by Bernard Helffer and Uzy Smilansky). Brian Davies presented an overview of recent developments in the spectral theory of non-self-adjoint operators, an exciting subject with lots of open questions. The talks of Vadim Kostrykin and Karsten Fritzsch focussed on applications of spectral theory to new problems arising in physics and engineering, such as the study of plasmons and metamaterials. Alexander Strohmaier discussed new results on precise numerical computations of spectral quantities on Riemann surfaces. Emily Dryden reported on her recent work lying on the interface of spectral theory and symplectic geometry. In several talks the use of numerical computation for creating mathematical conjectures was emphasized (talks by Brian Davies, Alexander Strohmaier, Bernard Helffer, Uzy Smilansky and Pedro Freitas). The talks presented at the workshop stimulated numerous fruitful interactions between the participants. The group included a large number of young researchers, in particular several Ph.D. students and postdocs, who benefitted from discussions with the renowned experts in the field. One of the highlights of the workshop was the open problem session chaired by Michiel van den Berg. A list of open problems formulated at this session and discussed during the Workshop can be found below. Geometric Aspects of Spectral Theory 2015 Workshop: Geometric Aspects of Spectral Theory