I first noticed this today when I sprinkled some Comet under the rim and it was immediately washed away by a little stream of water.
Does this mean we need a new toilet?
Yeah, I’m plumbing-impaired. So is my husband. He’s replaced toilet innards before, but water running from under the rim of the bowl is something new.
Thanks for any advice!
That just means the flush valve has a slow leak. You can probably fix it for five bucks or so by putting in a new “flapper”
It can also happen if the water level inside the tank is right at the top of the overflow tube. Take the top off the tank. There should be a fairly large vertical tube (1/2-3/4" diameter) in the middle of the tank. If the water level in the tank is at the top of the tube, then it’s draining into the bowl, and you need to fix the water level. You can do this by bending the metal rod that the float is on, or there may be a screw on top of the water valve which adjusts fill level.
If the water level is below the top of the tube, then the flapper is leaking, as gotpasswords said.
It could be that the tank overflow pipe is leaking due to the tank being filled too much. If it is that, adjust the filler valve or the tank float to stop the flow of water when it is full enough.
Thank you! My plumber guy is cute, but I really didn’t want to shell out for a new toilet. This sounds like an easy fix. 
I’m sure you don’t need a new toilet. But it’s important to know what you mean by rim. Or you referring to the bowl up top, or did you find water on the floor? If water on the floor, this too is a easy fix. Simply remove what is generally two nuts, and lift the toilet up, and put in a new wax ring seal. They only cost a couple of dollars.
The leak is from the top of the bowl, just under the rim.
I pushed down on the flapper and the water stopped running, so it looks like that’s the place to start. There wasn’t a metal rod to bend – this is one of those newfangled thingies, no float bulb, all plastic except for the overflow tube, which is metal.
Flappers rot over time. Luckily, toilet repairs are about the cheapest you will ever have to do. And the simplest. Before long you will be an expert and amaze your friends.
We’re assuming here the water is running inside the bowl, not outside, right?
If you’re using those large tablets of toilet cleaner that you’re supposed to drop into the tank every 2000 flushes, stop doing so. They rot your flapper at an accelerated rate.
Voyager, yeah, the water is running inside the bowl. Thank heaven. I know it’s easy to replace the wax ring but I’m not as young as I used to be, and toilets are heavy mothers.
Tom Tildrum, yeah again, we do use those drop-in things. I’ll stop.
Just take a good look at your flapper before you buy a new one - I think they come in two sizes. But otherwise, it’s so easy to fix that I didn’t call my landlord when the toilet was running - just fixed it myself. You can google instructions for replacement pretty easily. Also, when I replaced mine, the rubber was so old that it was turning into particulate - handling it briefly turned my hands black. It washes off, of course, I just didn’t want you to be alarmed if that happened to you.
dactylic hexameter, the plan is to remove the flapper and take it to the hardware store. Unless we do, we always come home with the wrong thing. You think you have a picture in your head, what the thing’s supposed to look like. With us, it’s always wrong, and we have a drawer full of unused toilet innards to prove it. 
There are also “one size fits all” flappers that are sealed by a ring of putty. They’re very useful if you have an old toilet (like I do).
I’ll check into the putty thing, thanks.
The leak has solved the mystery of why this toilet was the cat’s favorite. It has to be the running water.
I have one toilet where the flapper hole is of non-standard size. I used marine putty to fix it - with any luck you won’t need to. I’ve never seen flappers with putty, and my hardware store has a director’s chair with my name on it in the toilet aisle. (We’ve got three old toilets.)
Taking the flapper is an excellent idea. I learned it the hard way, luckily I only live a few blocks from the hardware store.
The marine putty goes around the hole in the tank? I’ll suggest this to my husband if we can’t find a flapper that fits.
Thanks again! You should be proud of your plumbing skills. Think of the money you’ve saved!
I believe it’s required by code that all plumbing projects need a minimum of two trips to the hardware store. Try as you might, it just can’t be done in one. You can bring the flapper with you, but it’ll end up being something else, you can plan on needing a three inch piece of copper but you’ll get the 1/2 inch instead of 3/4 inch. You can think you need a 6 foot long PVC pipe, so you buy one that’s 8 feet long just in case, but you’ll forget the PVC cement.
Joey P, the only reason I’m chuckling is that my husband is the one who has to make the two (or three) trips.
Not so with my first husband. He was a shade tree mechanic. He’d get started on something with the '58 Ford or the '59 Pontiac, and sure enough, he’d get halfway through the job and need a different size filter or gasket. His excuse was he was “too greasy” to go to Napa, so I’d make the trip.
The code applies to car maintenance too.