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Resistance gear is the major source of the Resistance stat. It has next to no general usefulness, and is usually used only in a raiding environment. Several bosses simply cannot be beaten without sufficient resistance of the appropriate type.
While leveling, resistance gear is usually not needed. There are some rare instances where it might be useful, but there hardly exists any low-level items with resistance. Prior to BC, the Molten Core was the first place which required high amounts of a resistance (Fire). Nearly every raid instance since then had one or more bosses which required some type of resistance - sometimes only on certain players (like the tank), sometimes on the whole raid.
Resistance equipment frequently has only two stats - a resistance and a high amount of stamina. In a wider sense, stamina can be interpreted as "resistance to all". In another sense, extra stamina while prolonging the time to die, does not mitigate damage as armor does. Using increased stamina in magic heavy fights, goes hand in hand with an increased healing requirement.
The problems with resistance gear are that it's rarely used, hard to find or manufacture, and wearing it increases survivability in a certain environment, but gimps most other character functions.
Some advantages of resistance gear are, the ability to ignore some encounter mechanics. During the air phase of the Sindragosa fight, hiding behind the ice tombs becomes less important with enough frost resistance, allowing more time to dps and heal, leading to more total damage or total healing. In some 10 player encounters, enough raid resistance will allow using 2 healers instead of 3, perhaps increasing raid dps. Resistance gear can lead to greater raid flexibility when a less that optimum mix of classes are in a raid.
This tradeoff, survivability vs. efficiency, always needs thorough consideration when choosing resistance gear. Use the best items available. Low-quality gear (like the green random BoE drops) is not really useful. It's much better to wear just one blue (or purple) item instead of a collection of greens. A similar example are warrior tanks using mail or leather resistance items instead of plate. Even in the most resistance intensive fights there is almost always some physical damage, and tanks should not nerf their physical mitigation too greatly. If there's no other choice such a measure might do the job, but the instances are designed such that good gear is available when a certain resistance is needed.
Farming marks (like [Badges of Justice]) or materials (e.g. [Heart of Darkness]) for appropriate resistance gear may still be an unavoidable part of raid progression.
Sapphiron requires the whole raid to have frost resist when first gearing up. As gear gets better, a paladin aura or shaman totem should suffice. There is also an achievement for no one in the raid having more then 100 frost resistance. [The Hundred Club (10 player)][The Hundred Club (25 player)]
Hodir requires the whole raid to have frost resist when at gear level of this instance. As gear gets better, a paladin aura or shaman totem should suffice.
It is unknown whether this raid will require frost resist as of the release of the Plagueworks.
Sindragosa does not require frost resist, but it is very helpful for the tanks.
Resistance & PvP
In theory, resistance could be quite useful in PvP. Unfortunately, most experienced PvP opponents wear large amounts of Spell penetration gear, which negates resistance. Therefore it's usually not a good idea to wear large amounts of resistance gear in PvP.
Knowing this, one might consider buffing stamina instead of resistance since stamina provides you with more hitpoints, therefore allowing you to soak more damage from both magical and physical attacks.