How do you capture live mice?

I have a mouse problem.  Possibly other approximately mouse sized animals.  The cat did bring in a chipmunk once, though I was able to get it out with the cat's (initially quite willing, later confused, finally semi-coerced) help.  Once we found a dead mole that was about mouse sized.  (Well ... closer to chipmunk sized)  For it to have been killed in the house it first had to be in the house.

But mostly mice, and certainly there are no signs of things significantly bigger than mice (the only rats that have ever been sighted in the house were pets, now all sadly departed.)

I have no desire to kill the mice.  I want them out of the house.  If the elements (or predators) kill the mice after I evict them then I'm not particularly happy with that outcome but they are wild animals and that's a risk that comes with being born free in a species without a civilization.

So I want to catch live mice.

All have a heart traps I know are for animals rather larger than mice.  As if mice aren't worthy of the same respect we give to lager animals.

I care about living things, except non-mammals that feed on mammal blood in a non-opportunistic way.  Which is to say that I have no great quarrel with vampire bats (though we totally need to find a way to give them mass inoculations so they won't be a disease vector) and neither do I mind things like butterflies that will drink blood if it's just sitting there, but don't actually make an effort to extract blood from a living host.

I don't want to hurt spiders and they're worlds more different from me than mice.

But letting the mice stay in the house is not good plan.  It's actually a very bad plan.  (Nevermind that it's the option we've been defaulting to since we moved here when I was less than one.)  The problem is worse now that I'm the only human in the house.  The cat is a part time employee at best and much prefers to hunt them in the less cluttered wilds.  The gecko stays in its cage.  I'm the only thing that's around often that the mice are afraid of, though they aren't that afraid of me, and I can only be in one place at a time.  (The Kwisatz Haderach I am not.)

This means that they grow ever bolder because big scary human thing is only ever in one part of one room.

So the mouse problem needs to be dealt with.

But the question remains: how the fuck do I catch live mice?  There's all sorts of technology for killing them, what is there for catching them.

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And as an aside, if I do catch them, is there anything that i can do to them which will make them deadly to fleas?

Because the flea problem only became the impossible to defeat flea problem after my sister brought over her dog and it had some kind of invasive super-fleas on it that spread to everything in the area.  (The squirrels never stopped and scratched themselves as if they were quite perturbed by being eaten by fleas before she brought her dog here.)

One assumes that, say, Frontlining some of the area mice might help things.  Of course Frontline doesn't come in mouse sizes and it would be a really bad idea to try to hold them still while you applied it.

Something that could be fed to them might work, but you'd still worry about overdose.

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But the main point is that I want to catch mice in my house without injuring them.  How the fuck does one do that?

[Added]

In case anyone was wondering, my house is not warm enough to simply buy snakes and let them loose.  If the fact that I wouldn't mind them being eaten by a predator I set loose while I don't want to hurt them in general seems like a contradiction ... snakes gotta eat.

Admittedly bringing a (non-cat*) predator in for the express purpose of killing mice isn't the same as putting them outside to deal with the predators there, or not, as they may.  But I think it's more a difference of degree than anything else.

* In my family house cats are an end in themselves.  That's why current one hasn't been fired due to her abysmal mouse record.