I'll add on answer despite one being already chosen since I want to add some things that I did not notice being mentioned and it doesn't make sense adding a "list of stuff nobody mentioned that I could see" as a comment to either a random answer or the question.
Already mentioned by others, but I noticed nobody mentioned why people transferred to using iron as soon as they had the technology.
Simply put copper alloys are too ductile. If used as weapon they have a tendency to deform, bend, and lose any sharpness the brownsmith tried to give them. This makes them pretty much useless for any weapon with a point or edge. And weapons with a length of metal such as swords and some polearms would have issues with bending.
On a positive note they are hard to break, which is why they were used for cannons and cartridges. Similarly they would probably (if cheaper due to the metals being more common) see use in armor. Bronze and brass actually were used in armor despite iron being available, if it was affordable.
First thing that occurred to me with some metals being more common than on Earth for some reason. If the area of your civilization was hit by a rain of iron-nickel meteorites, meteoric iron could be common enough to cause a fundamental change in metallurgy.
Why does it matter what kind of iron you use? Because meteoric iron is actually a nickel-iron alloy. It is found as pure alloy, no refining is needed. The nickel content also gives it much better corrosion resistance. Most alloys are also easier to work than iron, I think.
As a bonus it can have an exotic crystalline structure that looks pretty, might be magnetic (aka "magic") and can legitimately be said to be a gift from heavens.
A side-effect is that such abundance would probably be localized, so a specific area would probably have big edge on availability.
Also note that nickel is not that rare, so iron-nickel or cupronickel alloys might be possible solutions also. I doubt anyone would invent that without having metallurgy good enough for steel though.
There is also similar "telluric iron", if you dislike gifts from heavens.
Bamboo for example can be hardened to make decent knives and good spears. While metals are usually better you can actually make weapons and armor from wood.
And the superiority of metals is only an issue if your opponent actually has metal equipment. For building a setting it isn't that big a deal. As long as you remember the limitations that not having iron tools would cause for agriculture, masonry, carpentry, and the many other things that were transformed by access to cheap metal that keeps an edge.
You can in theory make weapons and armor from glass. It just doesn't make any sense if you have access to metals. The big issue is the brittleness. Glass has tendency to shatter on hard impact. This is not a good feature for a weapon or armor.
The simplest solution should be to laminate it with a material that prevents the cracks from spreading. Brass, bronze, copper, silver, gold, or their alloys should all work if the the layers are thin enough. Note that you do not need lots of the metal. Also note that actually manufacturing such a blade would almost certainly be a major pain. It really does not make any sense if you can use bulk metal instead.
Although a glass-gold laminate sword would be pretty cool... And would have the corrosion resistance to be a "sacred relic" for a very long time. Probably could outlast the species that created it. In a fantasy story a sword that was created at the dawn of time and has been sacred to multiple dominant species as they rise, flourish, and finally go extinct might be nice story item.
You'd probably want to make only the edge of the laminate and the rest of the weapon from wood in construction similar to what the Aztecs used with obsidian. Although the laminate would be more durable, so you could use a single blade instead of multiple microblades.
It never makes sense to make glass armor pre-modern since you can use wood, horn, hardened leather and cloth armor instead. The composite structure needed is just too complex without modern technology.
You can make weapons and armor from stone as there is really a wide variety of stone. Invariably they lose to metal in the ease of working them into durable forms. But if metal was unknown or unavailable and suitable stone was common it might happen. Or perhaps only swords made from jade can cut the demons and jade armor protect from their corruption...
Still stone can be used in combination with other materials to make effective weapons and armor. Essentially the other material supplies forms and resilience while small stones attached or embedded supply the hardness. Concrete is actually such a composite with different goals. Such weapons and armor are practical but require lots of work and maintenance.
As I mentioned before the Aztecs had a "sword" with small obsidian blades forming a cutting edge. It was fairly effective until it hit metal armor...
Of the above the meteoric iron is probably a usable answer. Bronze alloys, wood, and stone fall into the category of "good to know and might be of marginal use sometime". Like I said, this is kind of an extended comment that does not fit as a comment. The glass section is just a curiosity. It is probably too complex to be invented by a civilization that actually needs it. But if you want something really weird that comes from the distant past or a really exotic race...