Having dealt with a number of rats, this is a proven technique.
DO NOT use rat poison. The problem with rat poison is that it doesn't kill them instantly, so they can crawl back inside the wall and die there, leaving a horrible smell for weeks, until they decompose. Also rat poison is horrible to have around pets or kids.
Here is what you do:
1. Get rat traps, and a big jar of peanut butter. 2. On Day 1, lay the traps down without setting them, and put peanut butter on the trigger plate. 3. Repeat this for 2 more days without setting the traps, and only putting more peanut butter if they have eaten it. Wait until there has been 2-3 days in a row that they have eaten the peanut butter. They will think that they have a safe place to get food. 4. Apply the peanut butter, and set the trap this time. 5. Dispose of the rat and clean off the trap. Discard the trap if it has been blood-stained. 6. Repeat as needed.
The problem isn't so much catching them--peanut butter on a trap will eventually catch them. They also love to eat the packets of ketchup you get with French fries at fast food places. The real problem is getting rid of them--as in picking the dead thing up--and throwing it away. You're not going to put them in your trash can-right? That's what makes exterminators worth calling.
Here's what you do. Get a five gallon bucket and fill it 1/3 full of water. Drill 3/4" holes near the top so you can thread through a 3/4" or so rod. A large empty tin can can have holes drilled through each lid in such a way that it will spin quite freely on the rod. Cover the can in peanut butter and lay a ramp leading from the floor to the top of the bucket. Leave it far enough away from the can that the mice have to reach for it. They'll fall in the water when the can spins as they're attempting to get the peanut butter, and every so often you just remove the can, empty the bucket, and start over. I've caught ten mice in one night. And you don't have to risk touching the mice.
alka-seltzer tablets covered with peanut butter. As pointed out, rats can't burp. if you can get them to eat the alka seltzer they eventually die from internal injuries. Search the archives of Mother Earth News for details on the method.
Rats often won't take bait, so try the chocolate bait sandwich (if you are happy to use poison rat bait.) Line a plastic container with foil. Tip in a thin layer of melted chocolate. (Wearing gloves) scatter a sachet of poison over the chocolate.. Put another layer of chocolate on the top. When it sets you have a poison sandwich. Break it into pieces and store safely. Put two or three bits a night in a bait box away from kids and pets. The rats loved it so much they tried to drag off the bait box and I had to weight it down. Persist for two weeks - they will all eventually be killed. Rats take bait way. Mice eat it then and there.
Glue traps, poison, drowning, suffocating....all inhumane! Rodents are disease vectors and they of course do need to be removed from your home and property. No need to make them suffer though. RAT AND MOUSE TRAPS are the quickest and most humane way to get rid of them. After you trap one, wait a few days and set another one. Usually where there's one, there's two, sometimes three. After all of them have been eliminated, rodent proof your home so they will note want to return.