I've done a lot of research about mountain lions the last couple of years. Our house backs directly up to 10k+ acres of wilderness. Our property is in the home range of at least one very large cat. Last Summer there were tracks in our yard almost weekly. I have seen the animal three times and my wife has seen it twice. Each time we saw it slink off silently into the woods when it became aware of our presence. This is why it's still alive. I've talked to the local game warden many times about it and he's offered to issue me a "nuisance cougar removal permit"; set a trap in the yard that (he says) won't work; or come out with a team and hunt it down. At this point, I can shoot it at will without a permit as we have a well documented history of a problem with this animal. Washington State considers any big cat that frequents someone's yard a problem. If they get too familiar with humans it's just a matter of time before they start stalking them.
Female cougars here have home ranges of about five square miles that do not overlap with other females. Male ranges are larger and often overlap several females' ranges but do not overlap other males' ranges. All cougars will fight to the death to protect their home range against others of their same sex, including their own offspring. If a female kit doesn't leave it's mother's range at the "right" age she will kill it. So, when there's a population explosion where there is already "normal" population density the cats run out of range and start encroaching on human settlements.
As others have said, the cougar population has exploded in recent years. It's especially a problem here as hound hunting was outlawed so literally nobody "hunts" cougar. The warden said that about the same number of cougars are killed now as before the hound hunting ban, and by the same people as before. The hunters kept their hounds.
Now, instead of getting a cougar license they wait for a game warden to call them for removal of a nuisance animal. Then they get to go cougar hunting with their dogs. The cougar population here needs to be thinned, and thinning it at the edge of "civilization" probably makes more sense for us humans then guys hunting them way out in the wilderness.
If you support PETA don't read the following paragraph!
The basic way they remove problem cougars is: Three or four hunters/wardens with high powered rifles (plus side arms), and two or three teams of dogs track a cougar, tree it, and shoot it in the chest with a .22LR. They hope to hit it's lung(s) as this causes the animal to get very weak rapidly and not come down out of the tree after them. I know this sounds like a WAY cold way to kill an animal but you need to realize how dangerous a treed, wounded mountain lion really is. These cats will fight with everything they've got and they are brutal killers that live for the fight. Ever watched your house cat go after a mouse? Think about multiplying that by 50 or more and then wounding it! They don't just fight to their last breath, but until they lose consciousness.
Usually the cat falls out of the tree and as soon as it hits the ground they finish it with a hunting rifle. I'm sure if they were tracking one and saw it in open country they'd just drill it with a .308. Problem is there's not much open country here as the forests are choked with brush and this is very rugged terrain.
As far as handguns go, my local warden carries a .45 and has killed two charging cougars with it. He says they go down pretty hard with a hollow point to the chest but keep shooting until the gun is empty! Cougars are thin skinned and are similar in size and weight to humans so he says whatever works good on humans will work on cougars. I bought my .45 after my first conversation with him.... and my .308 after the second.
Another thing about mountain lions: If you are attacked fight back and don't ever give up! Protect your neck above all else as they have evolved with large canine teeth specifically for paralyzing their prey through spinal cord puncture. If it's got you down and you can get your hand in it's mouth do it. Grab it's tongue and pull or try to block it's airway. Then beat it to death with your other hand. This is assuming you can't put lead or steel into it of course.
I'm sorry if this post is incoherent. It's late but I wanted to relay what I've learned before the thread degenerated further and/or got locked.