The myth of the razor sharp sword
There are certain things that need to be extreamly sharp to do their jobs, such as a surgeon's scalpal or a razor blade and these items are designed with edge sharpness as their primary and practically only feature. In order to acheive and maintain this fine of an edge, the steel needs to be extreamly hard. Hardening the steel to this point will also make it very brittle, so only small blades can have edges like this. You can not simply scale up a razor blade to sword length, doing so would result in a sword so brittle it would shatter on impact like glass. To make a blade the length of a sword requires a series of compromises and carful balancing of properties. The blade needs to be hard enough to hold an edge, but not so hard that it can shatter. It needs to have some rigidity, but flex enough to absorb impact. Becasue a sword length blade can not be as hard as a razor blade, it will not hold the same edge as a razor blade. If you were to sharpen one up to that degree, you would find that it would dull itself very quickly. The combination of an extreamly thin edge without the extream brittle hardness will result in the edge rolling and folding over on itself very quickly. The end result is a blade that never stays very sharp because it was “over sharpened” in the first place. Think about how quickly a razor blade gets dull just from cutting whiskers. Now take a nesisarily softer piece of steel and subject it to significantly harder cutting materials and you get the idea.

Cutting paper, sure it looks cool, but...
This is something you have probably seen a lot. People like to show how sharp a blade is by cutting paper with it. With a knife, this can make some sense, some knives can be extreamly sharp, but with a sword, not so much. I have heard people criticize a sword saying that “it can not even cut paper” as if paper cutting is a prerequresite first step before being able to cut flesh. However, the truth is the opposite. When a sword blade is to the point of cutting paper it is getting past where it needs to be to function as a weapon and is closing in on becoming “over-sharpened” leading to an edge that will fold and not stay sharp. Being able to cut paper with a sword blade is about the pinacle of blade sharpness, not a starting point. Realistically, you should not be expecting most swords to ship quite this sharp for many of the reasons listed above.

So how sharp should a sword be?
A sword is simply a tool, a tool designed for the single purpose of killing other people, but a tool none the less. Like any tool it should be sharp enough to do it’s job and no more. Historically, a lot of swords were typically not as sharp as many people seem to assume. Many European Medieval swords were not super sharp. Some people say "butter knife" sharp, but think more like a kitchen knife that needs a touch up. The reasoning for this is simple, they did not need to be any sharper. Swords intended for battle would be given edges designed to survive as much battle as possible while still being able to perfrom their intended purpose. A sword does not require a super sharp edged to do it’s job, the combination of a moderate edge and force behind it is enough to split skulls and remove limbs. An edge that is super sharp does not actually perfrom these tasks any better in a real world situation, but will dull and chip faster. So swords were sharpened up to the degree that they would function without becoming too fragile. They were not sharpened up to impress other warriors by shaving hair off their arms or slicing pages out of illuminated manuscripts. The goal was to have enough “meat” on the blade edge that it would not be too damaged from an errant shot into a shield rim or plate armor.

If it's not sharp it's dull
People argue about this all the time, but I think a lot of the argueemnt stems from people using the terms differently. A lot of people seem to think of sharpness in black and white. "Razor sharp" or "dull". There is some logic to this, if your razor is not razor sharp, it would be considered dull. Knives are usually thought of the same way, once they are not super sharp, they are now "dull" and need to be resharpened. But swords are different. A sword with an edge like a dull razor blade would be considered "extreamly sharp". This is where I think people get mixed up. "sword sharp" is not really the same as "knife sharp", so "dull" on a knife can still be "sharp" on a sword. In essence a blade moves from sharp to dull when it stops being able to perform it's intended function. For a razor, that function is shaving, for a knife it could be filleting meat, but for a sword it can continue functioning at a far duller level by comparison with other types of bladed tools. Sharp is really not a word that can be used on it's own, the exact meaning of the term changes depending on what the sharp object is. Think of a "sharp" stick. The threshold of what makes a stick sharp can not even be compaired to what is required to make a razor blade "sharp".

So did they have sharp swords or not?
The short answer is yes, but we need to adjust our perception of "sharp" and remember to understand swords for what they are and not think of them as scaled up knives. The threshold of where a sword moves from dull to sharp is lower than a knife, do to the way it is used. To be clear, you can sharpen most sword blades up to "razor sharp". The question is if they should be that sharp and if historic ones ever were. Despite the history, some people like their swords that sharp and there is nothing really wrong with that. Just know that some sword blades will not hold that type of edge and others will become fragile. A sword that is "too sharp" will dull quickly and chip easier.

What about Katana?
The katana is considered by many the sharpest sword type in history. While most other sword types had to balance the opposing properties of hardness for edge retention with softness for flexability and shock absorbsion, the ingenious Japanese smiths developed a work around, actually a couple of work arounds. Blade lamination and diferential hardening. The Japanese were not the first to make laminated blades, but their process is important for our discussion of sharpness. Without going into great detail about lamination, you can think of the blade as a steel sandwich. The core or meat of the blade is made with a high carbon steel providing a very hard edge at the expense of being brittle. That hard core is then sandwiched with a softer lower carbon steel on the sides of the blade. The result is a blade with a very hard edge but a softer, more flexible spine. Expanding on this concept is differential hardening. In a normal blade hardening process the blade is heated to red hot and quickly quenched in water or oil. This will produce a very hard but somewhat brittle blade. With diferential hardening the back of the blade is insulated with clay. This way when the sword is quenced the edge cools quickly while the back cools slower. The result being a very hard edge with a softer spine.

The combination of these techniques along with the geometry of a katana edge can allow it to take and hold a sharper edge than can normally be acheived on a different sword type. However, the resulting edge can still be somewhat brittle, so the cautious Samurai would be selective on when he would use this prized katana and would likely choose to dispatch more enemeis, especially armored ones with the Yumi (bow) or Yari (spear). This leads us to talk about how little action swords actually saw through most of history compaired to other weapon types, but that's for another day.

So remember, when talking about sword edges, try to adjust your perception of "sharp" into sword mode and try not to think of it as just a big knife.