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I've stumbled into a conundrum in my 5e game. I think solving this question will help me.

What is meant by the rarity of the magic items if all magic items are rare? Does it mean that you can find more than one +1 longsword?

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This is explained in the Rarity section of Magic Items on page 135 of the DMG:

Common magic items, such as a potion of healing, are the most plentiful. Some legendary items, such as the Apparatus of Kwalish, are unique. [...] Rarity provides a rough measure of an item's power relative to other magic items. Each rarity corresponds to character level, as shown in the Magic Item Rarity table. A character doesn't typically find a rare magic item, for example, until around 5th level. [...] If your campaign allows for trade in magic items, rarity can also help you set prices for them.

Aside from being a rough measure of power, rarity, as somewhat stated above and by its very name, also provides a guideline for how frequently characters might find a certain type of magic item.

A legendary magic item is far more powerful and likely to be far more scarce than an common or uncommon magic item. Characters might find multiple Cloaks of Protection (uncommon), for example, throughout their adventures but would probably only ever find one Robe of the Archmagi.

Depending on the type of game your DM runs, obtaining magic items may be a rare occurrence but that still doesn't necessarily mean that all magic items are rare or that there's only ever only one of a magic item in existence.

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You asked: "Does it mean that you can find more than one +1 longsword?" which does not appear to be addressed in the other answers.

By the book, yes, you can find more than one +1 longsword. There is not just one +1 longsword.

Very few items are unique. "Some legendary items . . . are unique." (DMG p135). "An artifact is a unique magic item" (DMG 219).

On the other hand, the DMG considers a +1 weapon to be merely uncommon, and refers to +1 longswords almost generically: "for example, if a hobgoblin tribe has a +1 longsword" (p133), "a generic magic item, such as a +1 longsword" (p141) and also says that a 3rd level character could craft a +1 weapon (pp128-9).

So, while rated uncommon, it might be very likely to come across multiple +1 longswords in an adventuring career, or for several characters in a party to have one.

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Nearly all are too rare to buy, but some are more rare than others

Nearly all magic items are rare in the sense that you cannot just buy them, like you can mundane items, and in the sense that it also is difficult to create new ones (p. 135, DMG):

Unless you decide your campaign works otherwise, most magic items are so rare that they aren't available for purchase. Common items, such as a potion of healing, can be procured from an alchemist, herbalist, or spellcaster. [...] The game assumes that the secrets of creating the most powerful items arose centuries ago and were then gradually lost as a result of wars, cataclysms, and mishaps. Even uncommon items can't be easily created.

However, among those magic items that can be found in the game world, some are more rare than others:

Each magic item has a rarity: common, uncommon, rare, very rare, or legendary. Common magic items, such as a potion of healing, are the most plentiful. Some legendary items, such as the apparatus of Kwalish, are unique.

As you normally cannot buy them, in terms of game mechanics the rarity mostly is expressed by how hard it is to randomly find one adventuring. For this the game uses the treasure tables. In those tables, the more rare an item is, the later in the game (es expressed by PC level tier), you are likely to find one, and the fewer of that type you will find over the course of the game (same page):

Each rarity corresponds to character level, as shown in the Magic Item Rarity table. A character doesn't typically find a rare magic item, for example, until around 5th level.

How rare are the different rarities?

How this has been designed is explained in detail in Xanathar's Guide to Everything (funnily again on p. 135), which explains that conceptually there are major and minor items, and lists a detailed table showing how much of each rarity you find when.

tables in this section make a distinction between minor magic items and major magic items. This distinction exists in the Dungeon Master's Guide, yet those terms aren't used there. In that book, the minor items are those listed on Magic Item Tables A through E, and the major items are on Magic Item Tables F through I. As you can see from the Treasure Hoard tables in that book, major magic items are meant to be handed out much less frequently than minor items, even at higher levels of play.

Aggregated, this table looks like that for a whole party of adventurers:

Level/CR Common Uncommon Rare Very Rare Legendary
1-4 6 4 1 0 0
5-10 10 17 6 1 0
11-16 3 7 11 7 1
17+ 0 0 5 11 9
Total 19 28 23 19 11

About 80% of these are "minor" items, often consumables like potions or scrolls, and 20% are "major" items, often permanent, reusable items like magical weapons, staffs, or rings. If you assume a party of four, each character over their career would get maybe 2-3 legendary items, but about 7 uncommon items. So legendary items are much more rarely found than uncommon items.

That they are harder to find also reflects that there are not that many of them in the game world.

If you squint, there is an anomaly: common items are actually not the most common ones you find, uncommon items are. Common items are found about as often as rare ones. That is because you mostly find them in tier 1 while they do not make for exciting or meaningful loot for higher level adventures, and you spend a lot fewer levels in tier 1 than in tier 2 and 3; and also because in the core rules, those are the ones you can buy, like a healing potion, so there is less need to find them as treasure.

Power and Rarity

In addition, rarity is loosely associated with power (p. 135, DMG), but that relationship is pretty inexact. There are many items that are less rare, but more powerful, than other, similar items. For example, a broom of flying is more common than a potion of flying, or wings of flying, both of which are generally thought to be weaker. A ring of warmth is strictly more powerful than a ring of fire resistance, even though the latter is rarer. What is true in general is that on average, items tend to be more powerful the rarer they are, and most of the legendary ones are very powerful.

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Not all magic items are rare. The rarity of an item, as written in the DMG, determines how hard it is to find such an object. A potion of healing, for example is a magic item of a common rarity, which means a simple encounter for a level 1 group could have this in its loot. On the other hand some groups have to wait for 15 to 20 levels to get a legendary item, like the Archmagi Robe.

You can also have a look at the Treasure Tables, starting around page 235 in the DMG, to get a feeling on what encounter a party has to beat to get a certain item.

If your question was about magic items which have multiple rarities like the Instrument of the Bards or a Weapon +X, then that just means there are 'better' versions of the same item, that are harder to find. E.g. a longsword +1 can be found relatively early in a campaign, while a longsword +3 is more like an endgame weapon, since it has the best bonus one can achieve due to a weapon.

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  • \$\begingroup\$ You might want to use another item as the example, since you can buy healing potions based on the equipment list (PHB p. 150) for 50 gp. \$\endgroup\$ Commented Apr 2, 2016 at 15:06
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It really gets use in how soon to introduce items to characters. The rarer the item, the later in a characters life those level of rarity items should be introduced. It's really that simple.

It additionally marries with the cost of producing a magic item as well.

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