Since 1959, spaceflight, especially interplanetary spaceflight, has relied upon gravity assists to reach destinations all over the Solar System. Apart from Luna 3, all other examples we will consider involve using gravity assists to change orbits around the Sun, rather than some other object (such as the Earth in the case of Luna 3). The basic principle is as follows: changing orbits around the Sun requires changing spacecraft velocity relative to the Sun, known as heliocentric velocity. Conventionally, this is done using thrusters aboard the craft itself (e.g. with chemical rockets; see ion propulsion for another example). However, onboard thrusters always require ejecting mass, and heavier rockets are much more difficult and costly to launch.
Gravity assists take advantage of planets' orbital velocity and gravitational influence to alter a spacecraft's heliocentric velocity. Take the diagram below, which shows a probe flying by the planet Jupiter.
But this is not true: all velocities in the above diagram are relative to Jupiter, i.e. measuring the rate at which an observer on Jupiter would see the spacecraft traveling. What matters for its orbit though, is heliocentric velocity.
Suppose that, with respect to the Sun, our planet has an initial orbital velocity toward the left. Then the total heliocentric speed can increase for our spacecraft if the velocity relative to the planet is rotated to line up more closely with the planetary velocity (as indicated by the addition of arrows in the above diagram). Two of the greatest outer Solar System missions in history relied on this principle: the Voyagers.
Both Voyager 1 and 2 made use of a gravity assist at Jupiter to accelerate them to Saturn, and Voyager 2 did the same to reach Uranus and Neptune. While a slight change in direction at each planet is visible in the above diagram, the following graph better captures Voyager 2's changes in speed.
The blue curve graphs the magnitude of Voyager 2's heliocentric velocity at various distances from the Sun along its journey. Note that at each flyby of a giant planet, the probe's speed sharply increased for a short time (as it plunged into the planet's gravity well) but also was higher after each flyby than before, with the exception of after its final encounter with Neptune. The other curve shown is the Solar System escape velocity, that is, the velocity required at a specified distance from the Sun to ultimately escape its gravitational influence. Remarkably, Voyager 2 did not even have sufficient velocity to escape the Solar System until after its first flyby with Jupiter! Without gravity assists, the spacecraft would not be headed toward interstellar space today.
As indicated by Voyager 2's encounter with Neptune, gravitational assists can also reduce a spacecraft's heliocentric velocity. This is necessary for certain missions to the inner Solar System; objects launched from the Earth do not simply fall toward the Sun - they have to lose the angular momentum they inherit from our own planet!
In October 2018, the European Space Agency spacecraft BepiColombo on a mission to orbit Mercury in 2025. See here for an animation of the probe's seven-year trajectory. If the probe had been launched directly toward Mercury, the additional velocity acquired from falling into the Sun's gravity well would have made orbit impossible. Instead, the mission trajectory incorporated two Venus flybys and six Mercury flybys, all to slow down the spacecraft without the use of too much thrust.
Finally, some space missions even use encounters with planets to leave the plane of the Solar System! The Ulysses spacecraft, launched in 1990, had the goal of studying the Sun. In particular, it aimed to measure the solar wind (the flow of charged particles) and magnetic field emanating from the Sun. Unlike previous missions, Ulysses had the opportunity to study the Sun from above its poles in an orbit that was inclined 80.2° to the plane of the Solar System. A vast majority of solar system missions remain in the nearly flat plane of the Sun's equator in which the planets lie.
To achieve this unusual orbit, the probe went all the way to Jupiter just to flyby the giant planet. Jupiter's large mass allowed for a more effective gravity assist and its distance from the Sun meant that the probe was traveling slower there and the maneuver required a smaller change in velocity. Ultimately, Ulysses made a great deal of new discoveries, including that the solar magnetic field "flips" every 11 years.
Faced with the difficulties of efficient Solar System navigation, numerous space missions have utilized creative solutions involving gravity assists to reach their targets, providing another example of the spectacular innovations necessary to explore worlds beyond our own.
Sources: https://solarsystem.nasa.gov/missions/luna-03/in-depth/, https://solarsystem.nasa.gov/basics/primer/, https://www.researchgate.net/publication/228803791_Design_of_Lunar_Gravity_Assist_for_the_BepiColombo_Mission_to_Mercury, https://medium.com/teamindus/daring-gravity-assist-maneuvers-of-past-space-missions-411643cd3d55, https://solarsystem.nasa.gov/missions/ulysses/in-depth/
