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I noticed that weight is mentioned in race profiles and was wondering if that’s just fluff in the official materials or if it actually matters.

I might ask my DM if I can write up a system for his home brew world that makes PC weight matter.

Carrying another PC, speed reduction

Depending on which version of the encumbrance rules your table is using, if you end up in a situation where one PC has to carry another PC (for example, when fleeing a horde of creatures when you are low on resources) then the weight will allow the DM and the Player to determine the level of encumbrance, and thus movement penalty, that carrying the unconscious, paralyzed, sleeping, whatever PC levies on the carrying PC.

Lifting and Carrying (Page 63, Basic Rules).

Carrying Capacity.

Your carrying capacity is your Strength score multiplied by 15. This is the weight (in pounds) that you can carry, which is high enough that most characters don’t usually have to worry about it. Push, Drag, or Lift. You can push, drag, or lift a weight in pounds up to twice your carrying capacity (or 30 times your Strength score). While pushing or dragging weight in excess of your carrying
capacity, your speed drops to 5 feet.

Size and Strength. Larger creatures can bear more weight, whereas Tiny
creatures can carry less. For each size category above Medium, double
the creature’s carrying capacity and the amount it can push, drag, or
lift. For a Tiny creature, halve these weights.

A Goliath carrying a paralyzed gnome? Not a problem.

A halfling trying to carry a dwarf or another halfling? The added weight could slow them down considerably with a pack of wolves on their trail ...

Variant: Encumbrance (Basic Rules, p. 63)

If you carry weight in excess of 5 times your Strength score, you are
encumbered, which means your speed drops by 10 feet.

If you carry weight in excess of 10 times your Strength score, up to
your maximum carrying capacity, you are instead heavily encumbered,
which means your speed drops by 20 feet and you have disadvantage on
ability checks, attack rolls, and saving throws that use Strength,
Dexterity, or Constitution.

When using this rule, the disadvantage to things like ability checks (to climb up and down mountains while carrying a party member) could create a significant problem for the player to solve.

Can this mule carry us?

The carrying capacity of mounts is linked to weight; if the party is trying to get multiple wounded or incapacitated characters carried by a mount, their weight may create limitations. (PHB, p. 53-54) Similar situational considerations as mentioned above.

Can I pick up that gnome with my mage hand spell?

Depends on what the gnome weighs. Mage Hand is limited to 10 pounds. (thanks @mattdm)

How many characters can the wizard carry on their Tenser's Floating Disc?

TFD exploits and uses have a long history in D&D, but it has a weight limit (500 pounds).

Weight limits also apply to magic items like the Broom of Flying (slower flying speed if carrying more than 200 pounds), Rope of Climbing (3000 pounds), and Immovable Rod (8000 pounds).

Does this need homebrew?

I don't think so, but that's really a matter for your table to decide upon. A DM ruling with the above in mind should be able to handle most situations. (On the other hand, homebrew away if your table finds that fun).

Carrying loot and treasure, or the disabled character

If you find a bag of silver pieces ... or copper pieces ... or gold pieces ...

A standard coin weighs about a third of an ounce, so fifty coins weigh a pound

... but you are also being pursued by bandits, wolves, whomever you stole the coins from, etc ... how much the party can carry before being slowed down can become a decision point: how much treasure does one abandon to enable escape, or, with this much treasure after that long dungeon crawl, does the party fight (and if need be fight to the death) since it's the major haul?

This kind of decision has been part of the game since the 10 GP = 1 pound scheme in Original D&D three books published in 1974. It's can still arise as an issue, if the situation is that the party has a lot of heavy loot but needs to move quickly for an in-game reason. How hard your DM leans on this impact of weight, and the trade off between treasure and someone who needs to be carried, can result in some unlooked for role play and party interaction.

In some parties, one of the PCs may be unceremoniously dumped by the others who care for the loot more than their party member!

Yes, it can happen. But it doesn't have to. The strength of the bonds between party members informs this.

Similarly, if the "carried" character is an NPC, and the party drops the NPC to reduce weight, that too can lead to some interesting consequences and role playing.

Usually, weight does next to nothing

A PC's weight is largely a flavor thing that doesn't affect combat or other gameplay by default in the vast majority of cases. Being light or heavy has no effect for attack or damage rolls, fall damage, shoves or grapples, nor for the forced movement of spells such as thunderwave. In addition, none of the GMs I've played with haven't wanted to play up the importance of this little number that many people forget right after writing it down.

A character's weight can be notable when mounted or carried by another character or creature, as the carrying creature can become encumbered by the load imposed by the carried character. Some items like wagons also have carrying capacities, and may not be able to support the load of a too hefty PC. Magical items can also have load limits, such as the Immovable rod that supports weights up to eight thousand pounds.

Some adventures may play with PC weight

Without spoiling any particular campaign, some published adventures feature elements that can only support a particular amount of weight, or traps or mechanisms that activate once enough weight is placed upon a trigger. In cases like these, the PCs' weight can of course be very relevant --- lighter characters might avoid triggering a trap that heavier characters would.

Finally, apart from these your GM can use their own creativity to come up with any challenges or other set pieces that depend on your characters' weights, so this list is not exhaustive. Nothing limits the weight's significance apart from how much your GM wants to bring it up and how many ways to do this they can imagine.

Rules as Written, the specific weight is not often used

Most of the time, rather than worrying about the specific height or weight of creatures, the rules just abstract it using the creature sizes:

Characters of most races are Medium, a size category including creatures that are roughly 4 to 8 feet tall. Members of a few races are Small (between 2 and 4 feet tall), which means that certain rules of the game affect them differenlly. The most important of these rules is that Small characters have trouble wielding heavy weapons, as explained in chapler 6.

It doesn’t matter if you’re 4’ 2” or 7’ 5”, you’re still a Medium sized creature and anything that affects your size category also affects you. If a scything blade trap says it hits Medium sized creatures, you’ll be hit if you are between 4 and 8 foot tall, the specific height does not matter. Similarly, if a trap door opens when a Medium sized creature walks over it, your specific weight does not matter, only what size category you fit into matters.

However, @KorvinStarmast’s answer provides some exceptions to this where a character’s specific weight is important. The primary instance of where a character’s weight factors in to the game mechanics is when something is lifting the character or otherwise supporting or moving their mass. In these cases, it is important to know what the specific weight is.

If you want to make weight more important, here is my suggestion for how to do it

Taking inspiration from the creature size categories, you could introduce weight categories. Off the top of my head, I don’t know what the specific numbers would be, but essentially you would have categories that certain weights would fit into.

Such categories might be very light, light, medium, heavy, very heavy, extremely heavy (not the best names but you get the idea). There are 6 creature weight categories as there are 6 creature size categories. However, whilst the categories are related to each other, that doesn’t mean all medium creatures are medium weight or all small creatures are light weight. For example, although a Dwarf is a Medium sized creature, their weight may mean they fall into the “light” category. This means that any traps intended to be triggered by creatures of medium weight or heavier would not be triggered by a Dwarf. A particularly large or overweight Dwarf might push into the medium or heavy weight categories.

By introducing weight categories, you could easily make checks that revolve around weight without having to do unnecessary maths and bookkeeping (as you’d not only be calculating your PC’s weight but the weight of everything they are carrying as well). Personally, because of this question, I know plan to introduce weight categories into my games.