Algebraic Surfaces of General Type with K^2=2pg-1,pg >= 5

This paper mainly deals with minimal algebric surfaces of general type with K^2=2pg-1. We prove that for pg>= 7 all these surfaces are birational to a double cover of some rational surfaces, and all but a finite classes of them have a unique fibration of genus 2; then we study their structures by determining their branch loci and singular fibres. We study similarly for surfaces with strucutres by determining their branch loci and singular fibres. We study similarly for surfaces with pg=5,6. Lastly we show that when pg>= 13 all these surfaces are simply-connected.