  The main purpose of this survey is to provide an introduction,
algebro-topological in nature, to Hirzebuch-type inequalities for plane curve
arrangements in the complex projective plane. These inequalities gain more and
more interest due to their utility in many combinatorial problems related to
point or line arrangements in the plane. We would like to present a summary of
the technicalities and also some recent applications, for instance in the
context of the Weak Dirac Conjecture. We also advertise some open problems and
questions.
