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Basic Knowledge and Techniques
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 Conquerable Space
In EVE, systems can be divided into four different types: high security (0.5+ sec rating), low security (0.1-0.4 sec rating), NPC 0.0 and conquerable 0.0. Each is (on average) progressively more valuable in terms of the raw ISK production potential, due to better asteroids and better rats. Mercoxit, the rarest and most valuable ore, is unique to conquerable 0.0 and has a value per ore unit hundreds of times higher than that of Scordite, the best highsec ore. As such, having control of conquerable space is extremely lucrative.
The other main difference between conquerable 0.0 and the other types of space has to do with the stations. Unlike every other area of the game, stations in conquerable 0.0 are few and far between, meaning that logistics is much more difficult.
Although you can't actually conquer the space itself, what you can do is control the NPC stations within the space. A station is the only way to manufacture, the only way to access the market, and - short of a mothership - the only way to have a nearby clone. As such, when you control the station you effectively control the space.
 Capturing Conquerable Stations
Conquering a station held by another alliance, is unfortunately quite difficult. To capture a station, someone from the invading alliance must dock at the station and claim it as their own. However, to do this, you must bring down the station's shields. Aside from having a huge amount of hitpoints on the shield (in the neighborhood of 40 million), there is the matter of Player Owned Stations.
POSs are constructions players can put up at the warp-in point of a moon. In order to damage the shields of a system's station, you must have more POSs in the system than the defenders - and only the POSs of the largest size in the system are counted. So if the defenders have one large POS and three smalls up in the system, the invaders could put up twenty smalls of their own and it wouldn't make a difference - they have to have at least one large POS up there. Once the invaders have exceeded the defenders' POS count, they can then demolish the conquerable station's shields and dock, taking control of the station. However, the shields are still vulnerable to the defending side, until the invaders have kept more POSs of their own active in the system than the defenders for five downtimes in a row. At that point, the invaders gain Sovereignty of the system, and the conquerable station's shields become invulnerable.
That's complex. But it's the logistics needed for an assault of that sort that is staggering. A large POS tower alone, without guns or fuel, costs 300 million. And to attack a Large POS and destroy it with any kind of speed requires multiple Dreadnoughts, which apart from having massive skill requirements, cost two billion to buy and fit. POSs are also rather hard to take down.
Please see Sovereignty for more information.
 Assaulting Player Owned Stations
The theory behind assaulting a POS is this: you take your dreadnoughts in and pound on the POS until the shields reach 20%. At this point, if the owners have Strontium Clathrates (an expensive fuel with high volume) in the POS's hangar, the POS will drop into what is called Reinforced Mode. In Reinforced Mode, the station cannot be damaged. This mode continues, consuming 120 Strontium Clathrates per hour (for a large POS) until it runs out, at which point either the attackers can knock the shields all the way down and then destroy the station, or the defenders somehow stave off the attacking fleet while they use Ospreys to help recharge the POS's shields to 50%, at which point they can insert more Strontium.
Unfortunately, dreadnoughts are rather vulnerable while attacking a station, as going into Siege Mode (which boosts their damage and tank by a huge amount) immobilizes them. In addition, it only takes a small group of battleships to take down a dreadnought, and due to the dreadnought's horrendous tracking while in Siege Mode, it is unable to counter them on its own. So as well as a small dreadnought fleet, the dreadnoughts have to be protected by the attacker's battleship fleet, and the battleship fleet supported in terms of ammunition, tacklers, repair, et cetera.
This is why an alliance war is a huge undertaking, and why only a handful of alliances control the majority of conquerable space in EVE. Hundreds of veteran pilots are required, billions of ISK liquid and tens of billions of ISK in assets, along with a leadership who can make all this happen.
This is the "endgame" of EVE though, and will let you see things you wouldn't find anywhere else in the game - things like Dual Gigabeams, POS Sieges and warping around in a two hundred man fleet attacking another two hundred man fleet. It also paves the way to making insane amounts of ISK, be it via ratting triple 1.5mil spawns, or mining Mercoxit, which has twice the ore value per unit of Crokite.