Transduction barrier

Transduction barriers were protective shields surrounding Gallifrey and the Time Lords. These barriers could also be raised around Time Lord space stations, and TARDISes were normally unable to pass through them. (AUDIO: Trial of the Valeyard)

It took the disciples of Omega four generations to install the barriers, which were powered by the Eye of Harmony. They kept Gallifrey slightly out of phase with the rest of the universe, protecting the world from invasion by expelling anything foreign to Gallifrey, normally destroying the invasive force. There were at least five levels of intensity to the barrier. (AUDIO: Renaissance)

According to another account, there were four barriers. The First Barrier was a force field that could tear matter apart at a sub-atomic level. The Second Barrier was made of ideas, and refused to let anything pass because it didn't believe it was possible. The Third Barrier used reflection to counter any approach with an equal and opposite idea. The Fourth Barrier of Stubborn Philosophy refused to believe in the existence of any alien craft and would reiterate this until the enemy gave up. (PROSE: Doctor Who and the Krikkitmen)

Breaches
The Doctor's TARDIS was able to slip through the transduction barrier when the First Doctor stole it; he theorised that this was because the ship was deregistered. (AUDIO: The Beginning)

The barrier was deactivated by K9 Mark I on the orders of the Fourth Doctor, and the Doctor put a hole in the quantum force field above the Citadel. This allowed the Vardans and Sontarans to invade Gallifrey. (TV: The Invasion of Time)

Using Gallifreyan technology from the First Great War TARDIS, the Vast and Terrible Krikkit Fleet was able to simply switch the transduction barriers off. Romana diverted the energy from the barriers to the planetary rotation, moving Gallifrey out of the way. (PROSE: Doctor Who and the Krikkitmen)

One TARDIS did break through, but had to jettison nine tenths of its interior, and most of its circuits fused while its recall circuit burned away, leaving only the telepathic circuits functioning. (AUDIO: Prisoners of Fate)

A squad of Daleks which infiltrated Gallifrey lowered the transduction barrier, allowing them access to Gallifrey to invade. (AUDIO: The Apocalypse Element)

The Time Lord's vault on Tersurus was protected by an outdated transduction barrier which the Master was able to break through using Goth's TARDIS. Before the barrier could be updated and the break-in retroactively undone, the Master killed the vault operators. (AUDIO: Dominion)

The Eighth Doctor breached the transduction barriers to land the TARDIS outside Madame President Romana II's presidential palace. (WC: Shada) The War Doctor did it again during the Last Great Time War to catch the Time Lords' attention about his return to his home planet. (PROSE: Engines of War)

As he tracked the origin of a mysterious white energy that was consuming space, the Eleventh Doctor sailed down a micro-scale dimensional conduit, emerging on ancient Gallifrey, Knowing that this was supposed to be impossible, he suspected that the conduit was what had allowed his TARDIS to bypass the transduction barriers. (COMIC: The Lost Dimension)

The transduction barrier was still in effect following Gallifrey's relocation at the end of the Time War. Finding himself unable to get through it, and with no answer from the High Council, the Twelfth Doctor visited Karn to use the Sisterhood of Karn's secret door to the Capitol. (COMIC: Supremacy of the Cybermen)

Other information
Anti-time was unable to pass through the transduction barriers. (AUDIO: Neverland)

Bennus noted that none of Gallifrey's defences, including the transduction barrier, would be able to withstand the impact of the Thousand Worlds which the Daleks planned to forcibly crash into Gallifrey at fifty times the speed of light. (AUDIO: The Heart of the Battle)

Alternate timeline
In an alternate timeline, the Knights of Velyshaa breached the barrier and invaded Gallifrey. (AUDIO: The Sirens of Time)