From Wikipedia:
Rhode Island is a hybrid "shall issue" and "may issue" state for carry. Licenses may be granted either by local authorities or by the state's attorney general's office. Licenses granted by local authorities are "shall issue" while those issued by the attorney general's office are "may issue" under state law. Until recently, most local authorities had been deferring to the attorney general which effectively blocks most issuance, unless one is a retired law enforcement officer.
The practice of not issuing permits on a true "shall issue" basis has been the subject of recent litigation. In April 2015, the Rhode Island Supreme Court ruled that a police chief must accept and review carry permit applications and must render a decision and the reasons for that decision. More significantly, the court ruled that the issuing authority must "show cause" for denying an applicant a carry license.[2]
Permits issued by local authorities and the Attorney General's office are valid for concealed carry statewide.
Delaware is officially a "may issue" state for concealed carry, but mostly shall-issue in practice. Permits are generally issued to all applicants not barred from owning a firearm.
Open carry without permit is generally lawful. A 2014 Delaware Supreme Court ruling recognized that open carry was a long standing fundamental right, and could only be prohibited by local ordinances in effect prior to July 4, 1985. The city of Dover formerly required a permit from the police chief for a state concealed permit to open carry, but this was repealed in October 2015 in accordance with the ruling.