The winner will be decided by around 200,000 Conservative members, who will vote on the final two candidates over the summer.
Ex-chancellor Sunak has been backed by more MPs than Truss, but the foreign secretary is polling ahead of him with party members.
You have probably heard of the Conservative Party also being described as the "Tory party" in political coverage – but where does that name actually come from and what does it mean?
Where does the word 'Tory' come from?
It originally comes from the Irish Gaelic word tóraidhe, which means outlaw, robber or brigand, and became a term of abuse for an Irish rebel.
It's relation to the modern-day Conservative Party has been around for centuries, and the Tory Party originated during the reign of Charles II, specifically during the “Exclusion Crisis” from 1679 to 1681.
Parliament was split (sound familiar?) between the "Whigs", who wanted to exclude Charles’s brother, James, from becoming king because he was Roman Catholic, and the "Tories" who supported his claim to the throne.
The Tories won, and James became King James II.
Initially, the Tories started supporting a Catholic heir's rights to succession for the monarchy but later went on to be linked with Anglicanism.
The Conservative Party was renamed the Conservative and Unionist Party in 1912 after they had allied with the Liberal Union party who opposed the Liberal Party's stance on the Irish Home Rule.
The Irish Home Rule was a movement that campaigned for self-government for Ireland within the United Kingdom and Ireland which the Conservative and Unionist Party opposed.
However in 1922, Ireland's independence was granted leading to the renaming of the United Kingdom and Northern Ireland in 1927.
Under Sir Robert Peel, the term 'Conservative' in reference to the party was officially adopted in 1834. Peel was a two-time prime minister and founder of the Conservative Party. He modernised the party and reconfigured the key parts of old Tory tradition, claiming support for social reform and free trade.
However, despite the introduction of the new party name, 'Tory' stuck. The two terms became interchangeable and the older name was used in official titles such as Lord Hailsham's 'Toryism and Tomorrow' 1957 lecture.
Although the party was founded in 1834, it had no written constitution until 1998.
The evolution of both words mirrors the great adaptability of the party, which is thought to be its greatest strength and reason for its enduring hold on power.
How is the term Tory used today?
The term Tory tends to be more derogatory in its usage than Conservative, and practically speaking it's also a much shorter word. This is particularly useful for journalists writing headlines.
Political rivals tend to use the term Tory when attacking the party, for example referencing Tory austerity or Tory cuts.
As Britain’s oldest political party, Tory precedes the official Conservative name, but is still very much in use today.