For visits of up to 90 days visas are not required for most Western nationals, including Americans, citizens of EU countries, Australians, Canadians, New Zealanders, Malaysians, Singaporeans and most South Americans.

Residents of other countries should check requirements with a Swedish or Danish embassy.
When a visa is required, only one visa (either Swedish or Danish) is needed to visit both countries. For instance, the holder of a Swedish visa has the right to arrive via Copenhagen Airport without breaking any visa regulations, and vice versa. 

Applications for visas (stays of less than 3 months), residence permits (stays of more than 3 months) or work permits have to be submitted to a Danish or Swedish mission abroad, i.e. embassies and certain consulates general. 

Denmark abroad

Sweden abroad

Click here for more information about rules and regulations for visas and work permits