The development of ferrous metallurgy is a fascinating topic.

The earliest iron age weapons we have are indeed wrought iron and are from around the 1000 BCE. In Europe specifically we see iron weapons around 600bce.

The next big shift came in the transition from using bloomery iron, to blast furnaces which produced "pig" iron.

Bloomery iron was used to produce the early wrought iron, and was still in use in the 15th century CE, and was so long lived because it was possible to produce even with relatively low temperature furnaces. Because bloomery iron is made from iron oxide, it does not require the high temperatures needed to liquify steel or iron, which allowed for early furnaces to produce iron back before Europeans could produce furnaces that could liquify steel.

There are several remaining bloomery furnaces, including a 12 century example in Lapphyttan, Sweden.

The bloom that is produced in the bloomery process is a spongy mix of iron, silicates and slag, and needs to be forged and consolidated to get a solid piece of iron. Not only does this solidify and purify the porous iron into a uniform block, but the repeated forging a heating cycles carburise the steel, and folding it to further consolidate the steel - much like wrought iron - can distribute this carbon evenly. It is in this way that steel was produced first in Europe, by combining bloomery steel with wrought iron consolidation methods in a carburising forge.

The next big development was the blast furnace, around the 14th and 15th century CE. Some early examples date to the 12th and 13th century (moshyttan), but the processes was widely adopted in Europe around the 15th.

Blast furnaces, as the name implies, used forced air to smelt iron. The preheated air was forced into iron ore, charcoal and a limestone flux. The end product was pig iron, around 2-4 % carbon. In contrast to the bloomery process which produced low carbon content, the blast furnace produced steel with too much carbon. After consolidation, pig iron needed to be decarburised and worked to reduce the carbon content into to desires 0.5-0.8% range for arms and armor.

Carburising and decarburising steel can be achieved by heating the steel to specific temperatures and either allowing carbon (from the burning charcoal) to be absorbed, or by heating it until the carbon in the steel begins to react with oxygen in the forges atmosphere, reducing the carbon content of the steel.

Blister steel essentially was a carburising method used en masse for turning wrought iron into steel. It was developed in the mid to late 16th century

Crucible steel was only produced in Europe around the 18th century, and is the only method of these which fully liquified the steel. It's typified by very low slag content and good purity. Some crucible steel was hypereutectic (very high carbon, >1%) and could form patterns during the forging process, if it had the right alloying elements. This was produced in the middle east, central Asia and indopersian from around 300CE to around 1750CE. I have a full answer on this topic on a thread from /blacksmith

The crucible steel produced in Europe from around 1750 onwards was a typical medium to high carbon steel, and was clean and homogenous.

It's worth noting that beyond the manufacturing process of the steel, many of the medieval swords (and indeed earlier swords such as "viking" swords) needed to be pattern welded from multiple bars of steel and iron in order to achieve the desirable balance of edge retention, and flexibility and shock absorbance. This is more relevant with bloomery iron than blast furnace iron, but was done with both