Bit of a controversial topic. I think in Western countries men feel somewhat obliged or that it's their duty to pay for the first date, but appreciate if the girl offers to split it.
If you've been dating for a while I think on an equality level it's better to either split the bill or take turns in paying, but this will ultimately depend on where/how you've been brought up etc. In Russia, the women expect the men to pay for absolutely everything!