The　main　purpose　of　this　paper　is　to　prove　the　well-posedness　of　the　two-dimensional　Boussinesq　equations　when　the　initial　vorticity　omega　(0)　a　L　(1)(R　(2))　(or　the　finite　Radon　measure　space).　Using　the　stream　function　form　of　the　equations　and　the　Schauder　fixed-point　theorem　to　get　the　new　proof　of　these　results,　we　get　that　when　the　initial　vorticity　is　smooth,　there　exists　a　unique　classical　solutions　for　the　Cauchy　problem　of　the　two　dimensional　Boussinesq　equations.