Luveni was educated at Lautoka Fijian School and then at Nabua Secondary School in Suva, before enrolling in Adi Cakobau School in Sawani. She graduated in dentistry from the Fiji School of Medicine in 1967, the first Fijian woman to do so. After graduation, she worked for 20 years for the Ministry of Health, before working for the United Nations Population Fund as project manager for reproductive health from 1987 to 2002. She resigned three years ahead of the expiry of her term to take up a post as HIV project officer for the Ministry of Health, and now works as project manager for Fiji Network for People Living with HIV, a non-governmental organization.
The daughter of a former shopkeeper turned shipping magnate, Luveni hails from the village of Nukuni on the island of Ono-i-Lau, in the Lau archipelago, the sixth in a family of three daughters and six sons. With her husband, Lieutenant Colonel (ret.) Inoke Luveni, she has five children and six grandchildren. Her hobbies include tennis, softball, golf, netball, and volleyball.
As of 2008, Luveni is Minister for Health in the interim government headed by Prime Minister Commodore Voreqe Bainimarama. She is also co-chair of a government task team established to make recommendations regarding "social cultural identity and nation building" for the proposed People's Charter for Change, Peace and Progress.