Twenty-second Amendment to the United States Constitution
Some argue that the 22nd Amendment and 12th Amendment bar any two-term president from later serving as vice president as well as from succeeding to the presidency from any point in the presidential line of succession. Others contend that the original intent of the 12th Amendment concerns qualification for "service" (age, residence, and citizenship), while the 22nd Amendment, concerns qualifications for "election", and thus (strictly applying the text) a former two-term president is still eligible to "serve" as vice president (neither amendment restricts the number of times an individual can be elected to the vice presidency), and then succeed to the presidency to serve out the balance of the term (though prohibited from running for election to an additional term).
Marco Rubio
Rubio successfully ran for the United States Senate in 2010. In April 2015, Rubio announced that he would forego seeking re-election to the Senate to run for president. He suspended his campaign for president on March 15, 2016, after losing the Florida Republican primary to the eventual winner of the presidential election, Donald Trump. On June 22, 2016, Rubio reversed his decision not to seek re-election to the Senate; he went on to win a second term later that year.