There are a few options to prevent flat tires. Each have their pros and cons. Schwalbe Marathon Plus tires are a decent way to go.
A quick summary of the options is below for any others who wander through this thread. Flat tires are best avoided, but a basic knowledge of replacing and/or repairing a flat tire would also be advised.
These have a thicker layer of rubber on the outside rim, giving sharp objects more to go through. However, there isn't any practical difference between these and regular tires with regard to flats. If you go over something sharp, it will go through your tire regardless. Frankly not worth the money.
Liners consist of flexible Kevlar or nylon and are placed between the inner tube and the outside tire. These can be incorporated into the tire itself and sold as one piece (as with the Schwalbe Marathon Plus you mention and others). I use these personally, and haven't had a flat tire in over six years of trails and other harsh off-road use. These do add some weight to the tire, but only minimally add to rolling resistance. The feel and traction is comparable to a regular inner tube. These generally offer the best value.
Solid tires are never going to go 'flat' on you, as they are made of solid rubber. However, they tend to be very heavy (increasing rotational resistance), and don't always give you the same traction and feel as a properly-inflated inner tube.
Another recent fad was to load your inner tube with a 'goo' (Slime or other sealants). The idea behind this is that, as your wheel turned with biking, the inside surface of your inner tube would be coated with this substance. Any small puncture would be clogged and rapidly sealed by the escaping goo. This tended to weigh your tire unevenly as you bike, and you need to be aware of the way your tire is rotated when parking and re-inflating your tires (else the goo may clog the air valve too). It also doesn't work so well for larger holes, and it can make patch repairs messy.