The present publication contains a special collection of research and review articles on
deformations of surface singularities that put together serve as an introductory survey of
results and methods of the theory as well as open problems and examples. The aim is to collect
material that will help mathematicians already working or wishing to work in this area to
deepen their insight and eliminate the technical barriers in this learning process.
Additionally we introduce some material which emphasizes the newly found relationship with the
theory of Stein fillings and symplectic geometry. This links two main theories of mathematics:
low dimensional topology and algebraic geometry.¿ The theory of normal surface singularities is
a distinguished part of analytic or algebraic geometry with several important results its own
technical machinery and several open problems. Recently several connections were established
with low dimensional topology symplectic geometry and theory of Stein fillings. This created
an intense mathematical activity with spectacular bridges between the two areas. The theory of
deformation of singularities is the key object in these connections.
