Stout was born in the town of Lerwick, in Scotland's Shetland Islands. He retained a strong attachment to the Shetland Islands throughout his life. He was given a good education, and eventually qualified as a teacher. He also qualified as a surveyor in 1860. He became highly interested in politics through his extended family, which often met to discuss and debate political issues of the day. Stout was exposed to many different political philosophies during his youth.

In 1863, Stout emigrated to Dunedin, New Zealand. Once there, Stout quickly became involved in political debate, which he greatly enjoyed. He also became active in the Freethought circles of the city. After failing to find employment as a surveyor on the Otago gold fields, Stout returned to education, holding a number of senior teaching positions at the high school level.

Eventually, however, Stout moved away from education and entered the legal profession. In 1867, Stout was working in the law firm of William Downie Stewart, Sr. (father of the William Downie Stewart, Jr. who later became Minister of Finance). He was called to the Bar in 1871, and proved to be a highly successful trial lawyer. He also became one of Otago University's first students (possibly the first, although this claim is disputed), studying political economy and the theory of morality. He later became the university's first law lecturer.

Stout's political career started when he was elected to the Otago Provincial Council. During his time on the Council, he impressed many people with both his energy and his rhetorical skill, although others found him to be abrasive, and complained about his lack of respect for those who held different views.

Stout successfully contested an August 1875 by-election in the Caversham electorate and thus became a Member of the New Zealand Parliament. He unsuccessfully opposed moves by the central government (Vogel) to abolish the provinces. At the 1875 – 76 election a few months later, he was returned in the City of Dunedin electorate.

On 13 March 1878, Stout became Attorney General in the government of Premier George Grey. He was involved in a number of significant pieces of legislation while in this role. On 25 July 1878, Stout was given the additional role of Minister of Lands and Immigration. A strong advocate of land reform, Stout worked towards the goal of state ownership of land, which would then be leased to individual farmers. He often expressed fears that private ownership would lead to the sort of "powerful landlord class" that existed in Britain.

On 25 June 1879, however, Stout resigned both from cabinet and from parliament, citing the need to focus on his law practice. His partner in the practice was growing increasingly ill, and the success of his firm was important to the welfare of both Stout and his family. Throughout his career, Stout found the cost of participating in politics a serious worry. His legal career, however, was probably not the only contributing factor to his resignation, with a falling out between Stout and George Grey having occurred shortly beforehand.

At around this time, Stout also developed a friendship with John Ballance, who had also resigned from Grey's cabinet after a dispute. Stout and Ballance shared many of the same political views. During his absence from parliament, Stout began to form ideas about political parties in New Zealand, believing in the need for a united liberal front. He eventually concluded, however, that parliament was too fragmented for any real political parties to be established.

In the election of 1884, Stout re-entered parliament, and attempted to rally the various liberal leaning MPs behind him. Stout promptly formed an alliance with Julius Vogel, a former premier - this surprised many observers, because although Vogel shared Stout's progressive social views, the two had frequently clashed over economic policy and the future of the provincial governments. Many believed that Vogel was the dominant partner in the alliance.

In August 1884, only a month after returning to parliament, Stout successfully passed a vote of no confidence in the conservative Harry Atkinson, and assumed the premiership. Julius Vogel was made treasurer, thereby gaining a considerable measure of power in the administration. Stout's new government lasted less than two weeks, however, with Atkinson managing to pass his own vote of no confidence against Stout. Atkinson himself, however, failed to establish a government, and was removed by yet another vote of no confidence. Stout and Vogel returned to power once again.

Stout's second government lasted considerably longer than his first. Its primary achievements were the reform of the civil service and a program to increase the number of secondary schools in the country. It also organised the construction of the Midland railway line between Canterbury and the West Coast. The economy, however, did not prosper, with all attempts to pull it out of depression failing. In the 1887 election, Stout himself lost his seat in parliament to James Allen by twenty - nine votes, thereby ending his premiership. Harry Atkinson, Stout's old rival, was able to form a new government after the election.

At this point, Stout decided to leave parliamentary politics altogether, and instead focus on other avenues for promoting liberal views. In particular, he was interested in resolving the growing labour disputes of the time. He was highly active in building consensus between the growing labour movement and the world of middle class liberalism.

During Stout's absence from politics, his old ally, John Ballance, had been continuing to fight in parliament. After the 1890 election, Ballance had gained enough support to topple Atkinson and take the premiership. Shortly afterwards, Ballance founded the Liberal Party, New Zealand's first real political party. Only a few years later, however, Ballance became seriously ill, and asked Stout to return to parliament and be his successor. Stout agreed, and Ballance died shortly thereafter.

Stout re-entered parliament after a winning a by-election in Inangahua on 8 June 1893. Ballance's deputy, Richard Seddon, had by this time assumed leadership of the party on the understanding that a full caucus vote would later be held. In the end, however, no vote was held. Stout, backed by those who considered Seddon too conservative, attempted to challenge this, but was ultimately unsuccessful. Many of Seddon's supporters believed that the progressive views of Ballance and Stout were too extreme for the New Zealand public.

Stout remained in the Liberal Party, but constantly voiced objections to Seddon's leadership. In addition to claiming that Seddon was betraying Ballance's original progressive ideals, Stout also claimed that Seddon was too autocratic in his style of rule. Ballance's idea of a united progressive front, Stout believed, had been subverted into nothing more than a vehicle for the conservative Seddon. Seddon defended himself against these charges by claiming that Stout was merely bitter about not gaining the leadership.

One of the last major campaigns that Stout participated in was the drive to grant voting rights to women. Stout had long been a supporter of this cause, having campaigned tirelessly for his own failed bill in 1878 and Julius Vogel's failed bill in 1887. He had also been highly active in the campaign to increase property rights for women, having been particularly concerned with the right of married women to keep property independently from their husbands.

John Ballance had been a supporter of women's suffrage, although his attempts to pass a bill had been blocked by the conservative Legislative Council (the now abolished upper house of Parliament). Seddon, however, was opposed, and many believed that the cause was now lost. However, a major initiative by suffragettes led by Kate Sheppard generated considerable support for women's suffrage, and Stout believed that a bill could be passed despite Seddon's objection. A group of progressive politicians, including Stout, successfully passed a women's suffrage bill in 1893 through both the lower and upper houses, with the upper house narrowly passing it after some members who had not been in favour changed their votes because of Seddon's attempts to "kill" the bill in the upper house.

Stout was also involved with the failing Walter Guthrie group of companies in Southland and Otago which had been supported by the Bank of New Zealand, and (according to Bourke) Seddon was prepared to conceal Stout’s involvement - provided Stout left politics.

In 1898 Stout retired from politics. He had represented the seats of Caversham in the 5^th parliament (1875), Dunedin East in the 6^th parliament (1875 – 79) and in the 9^th parliament (1884 – 87), Inangahua in the 11^th parliament (1893), and the City of Wellington in the 12^th and13^th parliaments (1893 – 98).

On 22 June 1899, he was appointed Chief Justice of New Zealand, and remained in this position until 31 January 1926. As of 2011, Stout was the last Chief Justice of New Zealand to have served in the Parliament of New Zealand.

While Chief Justice, Stout showed a particular interest in the rehabilitation of criminals, contrasting with the emphasis on punishment that prevailed at the time. He took a leading part in the consolidation of New Zealand statutes (completed in 1908), and was made a Privy Councillor in 1921. In the same year as his retirement, Stout was appointed to the Legislative Council, the last political office he would hold.

Stout also had a role of considerable importance in the development of the New Zealand university system. He had become a member of the Senate of the University of New Zealand in 1885, and remained so until 1930. From 1903 to 1923, he was the university's Chancellor. He was also prominent in Otago University from 1891 to 1898, serving on its Council. He played a very significant role in the founding of what is now Victoria University of Wellington - the strong connection between Victoria University and the Stout family is remembered by the university's Stout Research Centre and its Robert Stout Building.

In 1929, Stout became increasingly ill, and never recovered. On 19 July 1930 he died in Wellington. He had been made a K.C.M.G. in 1886.