The default support settings seem to work fairly well with overhangs, but something like a quarter sphere might benefit from a raft as well. You may have a fairly small area in contact with the bed, and if the print gets too high up, any time it accidentally catches provides far more torque on the base to dislodge the model.

Generally for complicated shapes, I'd look at whether you can split it in half to get two nice flat edge which can then be on the printbed, after which you can glue them together.

Ok.  This was a great suggestion.  Slic3r does indeed have a pretty handy auto-support build built right into the software.  However, on my first run, it printed the first two layers ok, but then made spaghetti with the next 30 or so unattended layers.  When I finally killed the print, all but one vertical support printed in a wiry mess.  I opened the Slic3r options and slowed the support material printing down from 45mm/s to 10mm/s to test out a slow extreme.  This again made spaghetti out of the support.  It actually seemed worse off.  If I had to guess, the support material is printing too thin to actually support itself (ironic, given the name).  Because the support material is printing in single thread lines, any tiny deviation in alignment spells disaster after a few passes.  My print bed, like pretty much everyone's, is ever so slightly warped when heated.  I think that is causing the extruder nozzle to apply too much pressure in sections of the support material, so when it gain significant height, it knocks the support around instead of neatly printing on top of it.  Any suggestions? 

Lastly, I noticed that I could actually get away with small overhangs without support material.  If an overhang only hangs about a centimeter or less at 90* or  greater, then it seems to still print just fine without support.

op7ical: You are right.  Quarter hemisphere.  I guess I'm geometry-challenged.

Generally for complicated shapes, I'd look at whether you can split it in half to get two nice flat edge which can then be on the printbed, after which you can glue them together.

If it's a quarter-hemisphere, you can probably print it upright, with one of the sliced edges acting as the base.  You should use a brim to help strengthen its grip on the platform.  The only part of this that might be hard is the last bit of overhang.  But it is worth trying.

Hope that helps.

I've made a few successful test prints of items such as cubes and medallions.  Both of those items lay flat on the print bed, so as long as the first layer sticks, I am usually all set thereafter.  Now that my machine is decently calibrated and tested, I want to start to design my own 3D models and export them to STL for print.  My question is, when designing odd shapes, am I restricted to always having designs that lay flat on the print bed, or can I design items that only touch the print bed in a few areas.  This is where I'm kind of stumbling over my description. 

Say I design a perfect quarter of a hollow sphere with appropriately thick walls.  Is it possible to 3D print?  My newbie understanding is that I would only be able to position it on its back with the center touching and the edges flared up, or the opposite side with the three corners touching and the center floating.  I hope this makes enough sense for people to understand me question.