Commodore Cornelius Vanderbilt was in his 79th year when he decided to make the gift that founded Vanderbilt University in the spring of 1873. The $1 million that he gave to endow and build the University was the Commodore's only major philanthropy. Methodist Bishop Holland N. McTyeire of Nashville, a cousin of the Commodore's young second wife, went to New York for medical treatment early in 1873 and spent time recovering in the Vanderbilt mansion. He won the Commodore's admiration and support for the project of building a university in the South that would "contribute to strengthening the ties which should exist between all sections of our common country."

McTyeire chose the site for the campus, supervised the construction of buildings and personally planted many of the trees that today make Vanderbilt a National Arboretum. At the outset, the University consisted of one Main Building (now Kirkland Hall), an astronomical observatory and houses for professors. Landon C. Garland was Vanderbilt's first Chancellor, serving from 1875 to 1893. He advised McTyeire in selecting the faculty, arranged the curriculum and set the policies of the University.

For the first 40 years of its existence, Vanderbilt was under the auspices of the Methodist Episcopal Church, South. The Vanderbilt Board of Trust severed its ties with the church in June 1914 as a result of a dispute with the bishops over who would appoint University trustees.

From the outset, Vanderbilt met two definitions of a university: It offered work in the liberal arts and sciences beyond the baccalaureate degree and it embraced several professional schools in addition to its college. James H. Kirkland, the longest serving Chancellor in University history (1893–1937), followed Chancellor Garland. He guided Vanderbilt to rebuild after a fire in 1905 that consumed the main building, which was renamed in Kirkland's honor, and all its contents. He also navigated the University through the separation from the Methodist Church. Notable advances in graduate studies were made under the third Chancellor, Oliver Cromwell Carmichael (1937–1946). He also created the Joint University Library, brought about by a coalition of Vanderbilt, Peabody College and Scarritt College.

Remarkable continuity has characterized the government of Vanderbilt. The original charter, issued in 1872, was amended in 1873 to make the legal name of the corporation "The Vanderbilt University." The charter has not been altered since.

The University is self-governing under a Board of Trust that, since the beginning, has elected its own members and officers. The University's general government is vested in the Board of Trust. The immediate government of the University is committed to the Chancellor, who is elected by the Board of Trust.

The original Vanderbilt campus consisted of 75 acres. By 1960, the campus had spread to about 260 acres of land. When George Peabody College for Teachers merged with Vanderbilt in 1979, about 53 acres were added.

Vanderbilt's student enrollment tended to double itself each 25 years during the first century of the University's history: 307 in the fall of 1875; 754 in 1900; 1,377 in 1925; 3,529 in 1950; 7,034 in 1975. In the fall of 1999 the enrollment was 10,127.

In the planning of Vanderbilt, the assumption seemed to be that it would be an all-male institution. Yet the board never enacted rules prohibiting women. At least one woman attended Vanderbilt classes every year from 1875 on. Most came to classes by courtesy of professors or as special or irregular (non-degree) students. From 1892 to 1901 women at Vanderbilt gained full legal equality except in one respect — access to dorms. In 1894 the faculty and board allowed women to compete for academic prizes. By 1897, four or five women entered with each freshman class. By 1913 the student body contained 78 women, or just more than 20 percent of the academic enrollment.

National recognition of the University's status came in 1949 with election of Vanderbilt to membership in the select Association of American Universities. In the 1950s Vanderbilt began to outgrow its provincial roots and to measure its achievements by national standards under the leadership of Chancellor Harvie Branscomb. By its 90th anniversary in 1963, Vanderbilt for the first time ranked in the top 20 private universities in the United States.

Vanderbilt continued to excel in research, and the number of University buildings more than doubled under the leadership of Chancellors Alexander Heard (1963–1982) and Joe B. Wyatt (1982–2000), only the fifth and sixth Chancellors in Vanderbilt's long and distinguished history (Gordon Gee was named Chancellor in 2000). Heard added three schools (Blair, the Owen Graduate School of Management and Peabody College) to the seven already existing and constructed three dozen buildings. During Wyatt's tenure, Vanderbilt has acquired or built one-third of the campus buildings and made great strides in diversity, volunteerism and technology.

Today Vanderbilt University is a private research university of 6,319 undergraduates and 4,566 graduate and professional students. The University comprises 10 schools, a public policy institute, a distinguished medical center and The Freedom Forum First Amendment Center. Vanderbilt offers undergraduate programs in the liberal arts and sciences, engineering, music, education and human development as well as a full range of graduate and professional degrees. Employing more than 2,000 full-time faculty, a part-time and clinical faculty of approximately 1,500 and a staff of more than 14,200, Vanderbilt is the largest private employer in Middle Tennessee and the second largest private employer in the state.

• Center for Constructive Approximation, Department of Mathematics, Vanderbilt University
• Department of Mathematics, Vanderbilt University