You will find all kinds of ideas about controlling moles and few, if any, really work. Understand that most likely you have one mole at work because a mole needs a fairly large territory to meet its needs and it will defend that territory to the death if need be. A lot of mole activity might be considered a good sign because that means there is a lot of their primary food source, earthworms, available.
Trapping is the most effective method of control. Poisoned baits have not been shown to be effective against moles and are known to be harmful to other forms of wildlife. "Grub" poisons are very popular but those poisons most likely also kill the earthworms although the manufacturers and sellers of those products will assure you they do not. There are specific target controls, Bacillus popilleae for example, but any broad spectrum poison will kill off many life forms, including us if not used with due care.
Perhaps the linked article form Ohio State University will be of some help.

Here is a link that might be useful: Mole Control

Thank you very much for the info & the link. You kind of drove home what I already feared - trapping them (and it looks like "trapping" in this case is code for killing) them seems to be the answer. I just hate the idea of it and I am not even sure that any of the top surface runs are being used right now - I caved in several and see no re-building at all. Maybe I will be REALLY lucky and they will all be very old tunnels in an old uncared for lawn. Or maybe the useless Schnauzers that I took outside and made a big show over pointing out to them the mole tunnels (only to be met with lazy house Schnauzers laying down in the lawn and taking a mid afternoon nap) scared away the moles?!? - And how is it that 2 dogs will knock over furniture to chase a mayfly in the house, and yet the little critters that they were bred to chase in the lawn evoke absolutely no reaction - at any rate, I am going to get top soil today and try to fill the dips in the lawn and hopefully I will realize that the moles have truly moved on to greener pastures - or in this case, lawns.

I really don't mind killing things that sting or bite (or that I will eat), but in general I hate killing things just for living in the same space in which I want to live.

glitternguns, I feel the same way about yard critters. I believe in co-habiting with anything that does no harm. We have a mole problem in our yard too (we just bought this house less than a year ago) and I just set out my first veggie garden. Does trapping them necessarily mean killing them? I was hoping to find a way to trap and then bring to a far away field and release. Is that something that can be done?

You cannot rehome a mole nor is there such a contraption as a live trap for moles.

Moles feed on a wide variety of insects and other small animals that reside in the soil. Their favored food, when available, are beetle grubs and worms. They will also enter an area simply to forage, but won't stick around very long unless there is plenty of available food.

You may want to make occasional walks around the yard to flatten the tunnels so that damage doesn't occur to the grass. No stomping....you don't want to compact the soil.

As you become more aware of how bad (or not) the population is, you'll be better able to decide what to do. Trapping might not seem so bad if mole conventions seem to be going on in your yard.

You might contact your local extension office to see if someone on sfaff can tell you how to scout for white grubs in your lawn. I'm sure that there's ample information on line, too.