The daughter of a former shopkeeper turned shipping magnate, Luveni comes from the village of Nukuni on the island of Ono-i-Lau, in the Lau archipelago. With her husband Inoke Luveni she has five children and seven grandchildren. Her hobbies include tennis, softball, golf, netball, and volleyball.
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 twenty 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, working as project manager for Fiji Network for People Living with HIV, a non-governmental organization.
In January 2008 Luveni was appointed Minister of Health and then as Minister of Social Welfare, Women and Poverty Alleviation in the interim government headed by Prime Minister Commodore Voreqe Bainimarama. She was 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.
In 2013 Luveni indicated that she would stand as part of Bainimarama's proposed political party. She was elected Chairperson of the Lau Provincial Council in April 2014, succeeding Filipe Bole, but after being announced as a parliamentary candidate for the FijiFirst party in July 2014, she resigned that post after only three months in office. She was succeeded by Ilisoni Taoba. In the 2014 election she received 2,296 votes and was elected to parliament as the 14th highst-polling FijiFirst candidate. She resigned her seat immediately following the election in order to become Speaker, the first Fijian woman to hold the position. Her place as an MP was taken by Laisenia Tuitubou. She was formally elected Speaker on October 6, 2014.
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