I've tried everything in the above answers. I detest killing. BUT, the only way I was able to make a difference in my garden was to get a flashlight and go out at night while slugs were out of their hiding spots and cut them in half. Hundreds every night for 3 or 4 nights and then it drastically began to reduce. I'd go out every other night, every third or fourth night. I left the poor things with their guts all over my lawn. Someone said that they didn't like being near their dead relatives. I think that is bunk. But my goodness if I didn't have to worry about slugs in a few weeks of this grisly task. It is quick and humane...enough. I also would recommend getting rid of big chunks of wood, rocks around the edges of your planting beds and tall weeds. These are their homes.
I once had two muscovi ducks that gobbled up slugs and snails. Sigh. Then they shat them out on my back doorstep and that was worse...grin, good luck!
For your garden you need to install sheets of plexiglass or metal roofing around the perimeter, a foot high. A foot deep to secure. Beer traps, slug bait none of those work at all. Diatomaceous earth sort of works but the best way I have found is to go out at night with a flash light and scissors and chop chop slugs languishing on the lawn. They love to go out on the lawn at night, I don't know why but they do. Chop chop. Do for 3 or 4 nights. Doesn't have to be in succession. You'll see great results. The best you can hope for.
A slick barrier is the next best.
This is if you have already removed the condos with neon signs saying 'HOMES FOR SLUGS next to easy food'...No rocks, discarded lumber, lumber pieces for dividing, plastic is the worst possible solution. Beer is like neon signage, not a great idea and certainly ineffective.
Slug bait is dangerous for cats and dogs and I have never seen any interest in slug bait by slugs. Remember the word 'bait' means to attract. Similar to those electric fly zappers. They attract flying insects up to 5 miles away just to find themselves zapped into a zillion pieces all over the picnic table with thousands of viruses. Nuts. You do not want to attract pests.
Nematodes I've not tried...beneficial nematodes. Possibly a great addition to chop chop nights. Let us know if you go that route.