I remember taking a course in "Rough Terrain Rescue." It was held on a tower of 4 thick "telephone" poles with 3 floors, a fixed ladder up through access holes to each floor and walls on 2 sides only. Bloody cold when I took the class! It was given by some of the local full-time firefighters. We did repelling, belaying, swapping ropes half way up, Stokes basket work, etc. Tough and fun!!
I got there a bit early so I stood with the pro's on top of the tower until the rest of the students show up. As one student pulls in one of the guys teaching the class turns to the next and asks "How can you tell the difference between a pro firefighter and a volunteer??" <img src="/images/graemlins/confused.gif" alt="" />

"When you've got a $1200 light bar on a $300 car!"
<img src="/images/graemlins/grin.gif" alt="" />

If you decide to go the rappel route that others have suggested (that would be my suggesstion too!), you really don't need to buy yourself a nice harness. You're trying to escape from a burning building, not go on a climbing expedition. Some flat webbing made into a "diaper sling" would work just fine. Get yourself a locking carabiner (or two regulars - oppose the gates), a figure 8, and some climbing rope and you're all set. By "climbing rope" I don't necessary mean top dollar kernmantle stuff. We used to use what was called "Goldline" for rappelling. Not very good for climbing (too stretchy and stiff) except maybe for top-roping, but OK for rappelling. Find yourself an old piece of retired firehose (ask at a fire station) and cut off a 5 foot piece of it. Use it as a protective sleeve over your rope where you hand it out the window over potentially rope-cutting clutter.

Or, just take the stairs if you can!

I know the parent post was half-joking, in case I sound too serious, but most people assume that the only reason to escape out the window is fire.

Anyone familiar with the Northridge or other big quake correct me, but I've read that in many apartment buildings, stairways are one of the structurally weakest parts of the building. I'm talking about the flimsy stairs that are "open" to the outside world, not something like a concrete fire stairwell in the interior of a building. If the stairs go, then a window might be your only way out--and back INTO your apartment. And to get back in, I would prefer an escape ladder to a rope (well, actually, I'd prefer an elevator back up to Bee's fourth floor apartment, but that's just me... <img src="/images/graemlins/wink.gif" alt="" /> )