We give families of examples of surfaces of general type X with pg=0, K2=1 double covered by surfaces T with pg=0, K2=2. In Chapter 2 we classify all such constructions with |π(T)|=8, giving 4-parameter families of surfaces X for which π(X)=Z2 and Z4. There is a complete description of surfaces with pg=0, K2=l, π=Z4 in [Rl]. There was one example S with H(S,Z)=Z2 in [0&P]. The most interesting construction is the one in Chapter 3, for which πX={l}. This answers negatively the following question "are all simply connected surfaces with pg=0 K2>0 rational" coming from Severi's conjecture. These constructions were motivated by Reid's conjecture that if a given fundamental group H occurs, there should be examples X=T/Z2 with π(X)=H. In the Appendix we give an alternative proof of a formula for the arithmetic genus of a quotient surface, based on a remark of Hirzebruch.