INTRODUCTION
Rendezvous and proximity operations have been a part of human space activities from almost the beginning of human spaceflight. These have included:
To these manned activities have now been added unmanned RPOs such as:
HISTORY
The first RPOs were carried out in the Gemini series of flights, and the first rendezvous was carried out in 1965 and the first docking in 1966 (Gemini 6-6A,& 7).

The Gemini flights were to test the procedures that would be carried out by the Apollo lunar missions from 1969 to 1972. Below, an Apollo Lunar Module undocks from the Service and Command Module.

The Space Shuttle was designed with manned RPOs in mind. The Hubble Space Telescope was a major beneficiary of this when is was found necessary to correct the primary mirror that was inadvertently manufactured with a spherical rather than a parabolic curvature.
Below the Space Shuttle uses it 'Canadarm' to carry out an operation with another satellite.

Recent unmanned RPOs have included:
These RPOs have involved at least two unmanned spacecraft> They have all occurred in orbital regions above normal manned space spaceflight altitudes (ie >500 km) and some have included non-government entities.
They could pose legal challenges as to liability and safety, and in particular they could be both a positive and a negative contribution to space sustainability.
RPO DETAILS
A rendezvous maneuvre usually begins with the approach craft in a lower 'catch-up' orbit to close the orbital phase angel with the target craft.
Repositioning fuel burns brings the approach craft to a one orbit closure maneuvre planned to give an apogee at the target craft orbits' semimajor axis. A terminal initiation burn is made at this point for a final rendezvous exactly one orbit later.
Final approach brings the approach craft to the target craft just as it reaches the target position.

One of the problems with orbital maneuvres in 3D space is that more than one orbit parameter changes as a result of a rocket burn. Changing a will also change e and vice versa.

Orbit changes are usually quite counterintuitive.

As stated before rendezvous and proximity operations have been conducted as part of spacecraft operations since the 1960s and are becoming an increasingly important part of civil, commercial and national security space activities. Most of these RPOs have been benign and not potential threats. However, there are now an increasing number of RPOs which are not benign and which potentially threaten some space activities. It is now becoming common to designate RPOs and cooperative or noncooperative, and the latter are becoming the subject of heightened security.
In geosynchronous orbit, satellites can change their position due to a number of reasons. When servicing a particular group of customers they will "station keep" at a fixed longitude. If the customer group changes they will make a shift to a different longitude. Or the customer may desire to obtain images from a different region of the Earth's surface.

One example is the Russian Luch (Olymp) satellite which stationed itself near the Athena-Fidus satellite in 2017-2018. It is speculated that the reason for this was to monitor transmissions to and from the Athena-Fidus satellite. This is shown in the longitudinal plot below.

KEEPING YOUR DISTANCE
A geosynchronous satellite requires a certain volume of space to conduct its station keeping operations. Various forces including gravity of a non-spherical Earth, the moon act to move a geosat from its intended orbit. Other non-gravitational forces such pressure from sunlight and the solar wind may do the same. A geosat is normally allowed to drift in response to these forces. However, when the drift becomes too great and the satellite moves to the limit of its nominal station keeping box, ground control will fire its thrusters to bring it back into a more central position. The size of the station keeping box is 0.3 degrees which is equivalent to a distance of ~250 km at geosync orbit.

A satellite's nominal position also has a virtual space around it to keep it an appropriate distance away from other satellites. The absolute minimum separation distance is generally taken to be no less than 10 km. Any closer distance is usually regarded as either foolhardy (by friendly powers) or potentially hostile in the case of adversarial powers.
Russia, China and USA now routinely put 'spy' satellites into geosynchronous orbit. These satellites move along the geosynchronous belt examining the activities of other satellites.
China has conducted multiple tests of technologies for close approach and rendezvous that could enable it to take hostile action against an adversarial satellite. One particular series of such satellites is the SJ or Shijian satellites. Shijian is the Chinese word for "practice". These satellites have demonstrated the ability to dock with uncooperative satellites and tow them out of orbit. A very useful ability to remove derelict satellites out of the way.
Russia has also launched several satellites into LEO and GEO that have demonstrated the ability to rendezvous with other space objects, and to do so after periods of dormancy.
The United Stated has conducted multiple tests of technologies for close approach and rendezvous in both LEO and GEO, along with tracking, targeting and HTK (Hit to Kill) intercept technologies.
The US geosynchronous current spy satellite series are the GSSAP satellites (shown below).

These GSSAP satellites usually travel along geosynchronous orbit in pairs. China has released images taken by its SJ-20 satellite showing a GSSAP satellite approaching it to within 10 km (below).

RESOLUTION LIMIT AT GEOSYNC ORBIT
The table below tells you the minimum separation at which you can resolve two close objects in geosynchronous orbit given the resolution of your optical system. eg: Even with good seeing the best resolution is typically 2 arseconds. For an azimuth of zero the geosat distance will be 37000 km. This gives a separation of 180 m for two objects to be resolved. Closer than this the objects will appear as one (albeit brighter) object.
RESOLUTION LIMIT [METRES]
Visual
Angle [distance to geosat(km)]
(arcsec) 36000 37000 38000 39000
1 175 179 184 189
2 349 359 368 378
3 524 538 553 567
4 698 718 737 756
5 873 897 921 945
6 1047 1076 1105 1134
7 1222 1256 1290 1324
8 1396 1435 1474 1513
9 1571 1614 1658 1702
10 1745 1794 1842 1891
CHINESE SJ-21 RPO IMAGED
The images below were taken by Arie Verveer on 15/16 Nov 2021.
Image (a) Sidereal tracking on and 10 second exposure. This shows a near vertical trail of SJ-21 moving in an approximate north-south direction.
Image (b) Tracking off -meaning the telescope was stationary. The exposure was 10 seconds and shows a shorter and much brighter trail of SJ-21.
Image (c) This is a magnified version of image (b) and shows 2 close satellites with rapid identical rotation rates of just under 2 seconds.

A 10 second exposure is a trail length of 150 arcseconds, both for the satellite in image (a) and the stars in image (b).
The extent of the frame is 10 arc minutes for both axes.
The crosstrack movement (declination) of the satellite (image (b) is approximately 28 arcminutes or 5 km. This means the satellites were moving at 500 m/s. The separation of the satellites is ~800 m.
This image was taken about one month after SJ-21 was launched together with a sub-satellite for "practice". At this time the main satellite has managed to spin up to match the rotation rate of the sub-satellite. A successful practice of an RPO.
Australian Space Academy