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The Market is a feature unlocked after your first conquest (the planet of Antirion), allowing you to trade resources with the rest of the universe. It is displayed at the bottom of your UI, next to the Shipyard. You need to build a Trade Hub in order to use it on a particular planet.
Goods sold on the market are those from active planet. Goods bought from the market will arrive to the active planet. 
The Market deals on Market Coins, which can only be obtained from the Market. In other words, you have to sell things before you can buy things. You can only use Market Coins in the Market. Current Market coins are shown on the top left of the screen, below Influence.
All the trades routes are established from your active planet to New Babilo. It is thus very useful to trade from Ishtar Gate (after you conquer it) or at least Orpheus. Make one of those your Hub Planet for better routing of resources.
When you buy a resource, a ship will be dispatched from New Babilo to your active planet, carrying the desired item.
When you sell a resource, one of the Cargo Fleets from the active planet will take the resource to New Babilo (NOT a ship from the Hub Fleet). Keep in mind that the game will not necessarily select the most efficient fleet for the amount of resources (it can send a Big Cargo fleet for a mere 1000 units), nor the fastest (it can send a Big Cargo fleet even if fleets of ZB-22 Transporters are available). From the release notes: "The market will choose fleets giving priority to cargo fleets and choosing the best fleet according to the value max_storage*speed"
You can see those fleets on the "Market Fleets" section on the Fleets screen. Your returning market fleets will be shown on "Travelling Fleets"
The travelling times for market fleets are affected by the general acceleration of the 10-minutes idle bonus. Thus, it is advantageous to trade first thing when you log into the game, as you can then make several trades in a short amount of real time.
Only goods that you are currently able to extract/produce are available for trade.
Not every good can be imported at first (Ammunition for example) because there are just no exporters. You can see that there are no exporters listed below the name of the good, and furthermore, the stockpile is 0.
However, you can export every good that you possess, even if there are no importers (your market fleet will take the goods to New Babilo and leave them there). This means the resources you sell do increase the stockpile of that resource at New Babilo, from where you can then buy them back (at a premium).
On the Market screen, you can see (next to each resource) three numbers, in the form: X/YYY(Z/s).
- X is the amount that New Babilo has stockpiled. You can only buy as much as there is on the stockpile.
- Y is the maximum you can sell. It is fixed for every time travel (it is only affected by your Technology Points).
- Z is apparently the speed at which new resources arrive to New Babilo from the exporters.
- Even when the speed is red (negative) there is a stockpike on New Babylo that does not seem to go down, putting into question if the "importers" are even taking resources from the stockpile.
- It is also not known the specific amounts the exporters and importers take, since you only get a global amount of resource flow
The prices fluctuate, but usually there are some goods that are much cheaper than others. The Market prices are only barely correlated with how difficult to obtain are the resources. So you can buy some resources far cheaper than if would cost you to produce from raw materials. The main example is circuits, which you can buy by selling silicon. In fact, selling silicon will be your main souce of income early on.
- From v39D release notes
Resource prices will change accordingly to availability and a random factor.
- If there is a low availability of some resource, but a high demand, the price will go up, otherwise will decrease.
- It follows the formula: basePrice*random_factor*static_factor*dinamic_factor.
- The random_factor will be in the range of 0.8-1.2 and will increase/decrease every time some export/import is done by other civilizations up to 0.02 each time (the value is choosen randomly).
- The static factor will be equal to 1+0.2(log(1+stock)-log(1+max_stock))/log(1+max_stock).
- The dinamic factor will be in the range of 1 to 2. Every time an importer civilization tries to buy resources from the market but there is none available, the dynamic factor will increase by a random amount up to 0.003. Every time whatever amount of that resource flows into the market, the dynamic factor will be resetted to 1.
- On the Armor Games version, and contrary to the rest of bottom game sections (Extraction, Production, Energy, Research buildings, Shipyard) the active planet cannot be changed from the Market. You must change it from one of the other sections, or the Map. Change of active planet works fine on Kongregate version.