We discuss key results and milestones achieved while studying particular families of Diophantine equations as well as touching on open problems. It will be necessary to pull mathemagical bunnies out of a hat for this talk!
Links:
[1] http://www.mpim-bonn.mpg.de/taxonomy/term/39
[2] http://www.mpim-bonn.mpg.de/node/3444
[3] http://www.mpim-bonn.mpg.de/asymptotic25