All of my toilets in the house have started to run intermittently. Is there something wrong with the plumbing or could it be that the flapper has gone bad on all the toilets at the same time?
You have to diagnose why the toilets are running, but given that multiple toilets have started running simultaneously, I'm guessing your water pressure is too high. This can be due to a falling PRV or a missing or failing expansion tank. See this answer for the steps to check an expansion tank. This answer may also be helpful. If you find your water pressure is well over 60psi (I believe) all the time, then you need a PRV (or to replace your existing one).
If checking your water pressure shows that it's never high, then it's time to diagnose the toilets. If the float valve is adjusted too high and allows the water to reach the overflow, then you may be losing water through the overflow and the toilet will cycle to makeup the lost water. I've also seen cases where the tube to refill the toilet was installed in such a way that it was below the water level and it eventually siphoned water through the fill valve into the overflow drain. The water would siphon down until the toilet ran to makeup for lost water. The most common reason is a bad seal on the flapper valve. Depending on the flapper design, this may be replaceable without disassembling the toilet tank. One last possibility is a failing valve on the toilet. Typically, the reason these fail is from high water pressure. But the symptom will be a high water level, well above where the float should shut it off. You may find the float submerged if this happens.
There's nothing wrong with the plumbing...the flappers do go bad eventually. I h ave an ongoing problem with my 2 toilets. Not that it's really on-going, it's that I put it off until I can't stand it.
Replace one of them (the flapper or the whole kit) and see if the problem is solved (it will be!). I don't have to tell you what to do with the rest of your plumbing if this works.
Remove the tank lid and look inside. There will be a copper tube sticking up in the middle. If the level of water is up to the top of the copper tube and draining into the tube, which would make it above the max fill line, you have a bad fill valve. The fill valve is the component that sticks up from the left side (usually) of the tank and has the float (or float arm is connected to it in the case of older style ball floats.)
If the water level is not up to the top of the copper tube you have a bad flapper that isn't making a good sel and letting water escape. The flapper is a rubber lid that covers the outflow port on the bottom middle of the tank. It is connected to the lever via a chain and arm.
You're best bet is to just replace them both to save yourself the hassle of replacing the other at a later point. They come in a kit. Take a picture of the flappers to make sure the one in the kit will work or get an appropriate flapper.
I really like the Korky QuietFill valves. They're quieter than the FluidMaster ones they replace and have lasted longer for me. I think the oldest one I have installed is going on 10 years now without problems.