Introduction
1 Notation
2 The classification problem and "simpler" sub-problems
2.1 Update on surfaces with pg=0
3 Other reasons why surfaces with pg=0 have been of interest in the last 30 years
3.1 Bloch's conjecture
3.2 Pluricanonical maps
3.3 Differential topology
4 Construction techniques
4.1 Quotients by a finite (resp.: infinite) group
4.2 Galois coverings and their deformations
5 Keum-Naie surfaces and primary Burniat surfaces