President Bouteflika, who is 81, is currently being treated at the Geneva University Hospitals in Switzerland and the protest movement is calling for him to step down.
There is nothing in the Algerian constitution to stop Mr Bouteflika seeking a fifth term.
"Bouteflika was sick when he sought a fourth term in 2014, but his medical results for the past five years were good," the prime minister, Ahmed Ouyahia, told parliament on Thursday, 28 February.
Mr Bouteflika has already won the endorsement of several political parties, trade unions and business groups and the opposition to the FLN are deeply divided.
So what is going on in Algeria? Here are five things about the country you may not know.
The Day of The Jackal
One of the biggest hit movies of the 1970s was set against the backdrop of Algeria's war of independence.
In 1971 Frederick Forsyth published The Day of The Jackal, a tense thriller which was followed two years later by the film, starring Edward Fox as the eponymous character, a hired assassin who is paid to kill French President Charles de Gaulle.
The book and the film begin with the failed assassination of de Gaulle in 1962 by the Organisation Armée Secrète (OAS), a group of die-hard French settlers who were furious at the former General for granting Algeria independence.
Jean-Marie Bastien-Thiry, who masterminded that attempt, was executed in 1963.
France had occupied Algeria since 1830, more than a million French settlers called it home and it was so integrated into the motherland that few thought Paris would ever grant independence.
But in 1954 an Algerian rebel movement, the FLN (Front de libération nationale) fought a guerrilla war against the French colonial authorities and eventually forced de Gaulle to concede defeat.
It was that "betrayal" which marked him out for an assassin's bullet. But de Gaulle died peacefully in 1970 and most of the French settlers, the pieds noirs, returned to the mainland.
The FLN has formed every government in Algeria since 1962.
When it did lose an election, in 1991, it was quickly annulled, triggering a decade-long civil war.
The Average Algerian is Only 27
The average age in Algeria in 2015 was 27.5, compared to 41.2 in France and 46.3 in Japan.
Algeria has seen huge demographic change over the last half a century.
When it became independent in 1962 the population was nine million and shortly after that it dropped as a million settlers returned to France.
But a baby boom followed independence and as mortality rates declined, the population began rising rapidly.
In the 1980s the average age was as low as 16.7 and by 2018 the population had reached 42 million.
Some journalists want to call on their colleagues from the public and private sectors to boycott Abdelaziz Bouteflika's activities, his outings and those of his electoral staff, discussions under way to see the feasibility of such an action #algeria— Khaled Drareni (@khaleddrareni) 1 March 2019
In recent years improved education about birth control for Algerian women has led to a fall in the birth rate and by 2050 the average age is predicted to rise to 37.
In the late 1980s, the "youth bulge" led to high unemployment, with many Algerians migrating to France. Today, around a quarter of young Algerians are unemployed; however, this is still much lower than in countries such as Spain (32 percent), for example.
Many young Algerians use social media and smartphones like their European counterparts and are pushing for an Arab Spring-style overthrow of the FLN regime, apparently unaware of the possible consequences, as seen in neighbouring and anarchic Libya.
Algeria is Sitting Pretty — on a Huge Gas Field
Algeria has the 10th largest proven reserves of natural gas in the world, is the sixth largest gas exporter and has the third largest proven reserves of shale gas — behind China and Argentina but ahead of the United States.
As a result Algeria has US$90 billion in foreign currency reserves and its external debt is only two percent of GDP. So it is not a poor country.
And there are vast areas beneath the Sahara Desert which remain unexplored.
In July this year the government is expected to pass a hydrocarbons law, which would reduce taxes for foreign companies, simplify bureaucracy and speed up the time it takes to get gas fields operating.
But gas operators have suffered attacks from Islamist extremists.
In January 2013 gunmen from al-Qaeda In The Maghreb (AQIM) attacked the In Amenas gas plant in southern Algeria, killing 40 workers, seven of them British nationals.
The Algerian civil war — which lasted from 1991 to 2002 — saw the death of 100,000 people, mainly at the hands of Islamic extremists from the Islamic Salvation Front (FIS).
Prime Minister Abdelmalek Sellal introduced a series of reforms in 2016 — designed to boost the agriculture, manufacturing, tourism, IT and renewable energy industries.
These have largely succeeded and Algeria has also attracted Renault and Volkswagen to set up factories in the country, which will supply the African market.
The government also recouped 40 billion dinars ($30 million) from tax dodgers and attracted a $3 billion Chinese investment in the port of El-Hamdania.
Mr Sellal's replacement, Ahmed Ouyahia, introduced further reforms which have been praised by the International Monetary Fund.
Africa's Most Expensive Footballer is Algerian
Football is hugely popular in Algeria and the best known player with Algerian ancestry is Manchester City Riyad Mahrez, who signed from Leicester for £60 million in July 2018.
Mahrez was born in Sarcelles, a suburb of Paris, but his father was Algerian and he chose his dad's nation over France and was called up for the Desert Foxes' squad for the 2014 World Cup.
The winger began his career with Quimper in northern France and, after a spell at Le Havre, joined Leicester with whom he won the English Premier League in 2016.
Title-winning manager Claudio Ranieri said he was "priceless" but eventually, when he made it clear to Leicester's owners he wanted to move on, they accepted £60 million for his services.
While living in Leicester, Mahrez met his wife Rita Johal, a model whose cleavage shots on Instagram would not go down well in many parts of Muslim Algeria.
Mahrez is a practising Muslim and went on the haj pilgrimage to Mecca in 2017.
Few outside Algeria or France will have heard of Rai music, but it is hugely popular, has produced dozens of star performers and has influenced everything from hip hop to classical music.
Most of rai's stars — Cheb Mami, Chaba Fadela, Cheb Sahraoui, Chaba Zahouania, Cheikha Rimitti, Houari Benchenet, Cheb Hamid, and Messaoud Bellemou — hail from the city of Oran in western Algeria.
Probably the biggest star of them all is Khaled.
Khaled — who turned 59 this week — was born in Oran's and began singing at the age of 14.
In 1986 he moved to France and cut his first album, then known as Cheb Khaled (Young Khaled).
His biggest single, Didi, in 1993 was a huge hit all over the Arabic-speaking world as well as in France and Belgium.
Singing in French or Arabic, the man known as the King of Rai has sold more than 80 million albums worldwide.
But rai singers like Khaled have often fallen foul of both the state and the imams.
Rai is sung in Oran's colloquial Arabic, which borrows words from Spanish, French and Berber, and singers like Khaled often sing about women and alcohol, which of course are frowned upon by Islamists.
"Khaled's directness and his force-of-nature voice…didn't sit so well with the growing number of Islamic fundamentalists in Algeria…and his songs were consequently banned from state radio," said Banning Eyre, a world music expert on NPR radio in the US.
In his 1998 autobiography, Derrière La Sourire (Behind The Smile), Khaled said he was invited onto a live TV show in Algeria in the early 1980s.
He was warned not to sing about sex or any other "vulgarities".
So he sang three songs — one about the Prophet Muhammad, one about love and one about alcohol and women.