The Brayton cycle, also known as the Joule cycle, is a thermodynamic cycle that describes the operation of certain heat engines that have air or some other gas as their working fluid. It is characterized by isentropic compression and expansion, and isobaric heat addition and rejection, though practical engines have adiabatic rather than isentropic steps.

The most common current application is in airbreathing jet engines and gas turbine engines.

The engine cycle is named after George Brayton (1830–1892), the American engineer, who developed the Brayton Ready Motor in 1872, using a piston compressor and piston expander. An engine using the cycle was originally proposed and patented by Englishman John Barber in 1791, using a reciprocating compressor and a turbine expander.

There are two main types of Brayton cycles: closed and open. In a closed cycle, the working gas stays inside the engine. Heat is introduced with a heat exchanger or external combustion and expelled with a heat exchanger. With the open cycle, air from the atmosphere is drawn in, goes through three steps of the cycle, and is expelled again to the atmosphere. Open cycles allow for internal combustion. Although the cycle is open, it is conventionally assumed for the purposes of thermodynamic analysis that the exhaust gases are reused in the intake, enabling analysis as a closed cycle.

In 1872, George Brayton applied for a patent for his "Ready Motor", a reciprocating heat engine operating on a gas power cycle. The engine was a two-stroke and produced power on every revolution. Brayton engines used a separate piston compressor and piston expander, with compressed air heated by internal fire as it entered the expander cylinder. The first versions of the Brayton engine were vapor engines which mixed fuel with air as it entered the compressor; town gas was used or a surface carburetor was also used for mobile operation . The fuel / air was contained in a reservoir / tank and then it was admitted to the expansion cylinder and burned. As the fuel/air mixture entered the expansion cylinder, it was ignited by a pilot flame. A screen was used to prevent the fire from entering or returning to the reservoir. In early versions of the engine, this screen sometimes failed and an explosion would occur. In 1874, Brayton solved the explosion problem by adding the fuel just prior to the expander cylinder. The engine now used heavier fuels such as kerosene and fuel oil. Ignition remained a pilot flame. Brayton produced and sold "Ready Motors" to perform a variety of tasks like water pumping, mill operation, running generators, and marine propulsion. The "Ready Motors" were produced from 1872 to sometime in the 1880s; several hundred such motors were likely produced during this time period. Brayton licensed the design to Simone in the UK. Many variations of the layout were used; some were single-acting and some were double-acting. Some had under walking beams; others had overhead walking beams. Both horizontal and vertical models were built. Sizes ranged from less than one to over 40 horsepower. Critics of the time claimed the engines ran smoothly and had a reasonable efficiency.

Brayton-cycle engines were some of the first internal combustion engines used for motive power. In 1875, John Holland used a Brayton engine to power the world's first self-propelled submarine (Holland boat #1). In 1879, a Brayton engine was used to power a second submarine, the Fenian Ram. John Philip Holland's submarines are preserved in the Paterson Museum in the Old Great Falls Historic District of Paterson, New Jersey.

In 1878, George B. Selden patented the first internal combustion automobile. Inspired by the internal combustion engine invented by Brayton displayed at the Centennial Exposition in Philadelphia in 1876, Selden patented a four-wheel car working on a smaller, lighter, multicylinder version. He then filed a series of amendments to his application which stretched out the legal process, resulting in a delay of 16 years before the patent was granted on November 5, 1895. In 1903, Selden sued Ford for patent infringement and Henry Ford fought the Selden patent until 1911. Selden had never actually produced a working car, so during the trial, two machines were constructed according to the patent drawings. Ford argued his cars used the four-stroke Alphonse Beau de Rochas cycle or Otto cycle and not the Brayton-cycle engine used in the Selden auto. Ford won the appeal of the original case.

In 1887, Brayton developed and patented a four-stroke direct-injection oil engine. The fuel system used a variable-quantity pump and liquid-fuel, high-pressure, spray-type injection. The liquid was forced through a spring-loaded, relief-type valve (injector) which caused the fuel to become divided into small droplets. Injection was timed to occur at or near the peak of the compression stroke. A platinum igniter provided the source of ignition. Brayton describes the invention as: “I have discovered that heavy oils can be mechanically converted into a finely divided condition within a firing portion of the cylinder, or in a communicating firing chamber.” Another part reads, “I have for the first time, so far as my knowledge extends, regulated speed by variably controlling the direct discharge of liquid fuel into the combustion chamber or cylinder into a finely divided condition highly favorable to immediate combustion.” This was likely the first engine to use a lean-burn system to regulate engine speed and output. In this manner, the engine fired on every power stroke and speed and output were controlled solely by the quantity of fuel injected.