Does an induction stove require flat bottomed vessels?
Most reputable sources say that curved surfaces such as woks don't work as well on induction stoves. They even make special tools and cooktops for inductively heating woks.  This phenomenum could be because of the angled surface or the extra distance from the cooktop, but it's probably both.  It's not that these surfaces are immune to induction heating, just that they aren't efficient.

Angled surfaces won't respond as well to induction stoves since the maximum amount of energy is generated in a surface when it's flat with respect to the stove.  (Magnetic induction is proportional to the cosine of the angle between loops, for those of you who want to know the physics).

Distance is probably the biggest factor.  Induction works at a small distance from the stove (there are numerous examples on the web of induction heaters working through newspapers or a magazine.)  However, the design of these cooktops makes it so the magnetic field that drives the heating dies off over a very short distance (For example, the Swiss Federal Office of Public Health determined when studying their effect on pacemakers that the field is negligible after 10 cm or 4 in).

As far as what materials work, usually cookware that's induction cooktop compatible has a curly icon on the box to indicate this.  Common compatible materials include magnetic stainless steels (not austenitic steels), iron, and carbon steel.  The article on induction cooking on Wikipedia has a great explanation on what cookware works for induction cooking and why.

There common rule of thumb is that if a magnet sticks to the bottom of a pot or pan, it's compatible with an induction stove. Surface resistance is a more scientific measure, but to get the  complete picture of why certain things do or don't work would require a college course in material electromagnetics. For a more detailed analysis, see the wikipedia page I linked. For most cookware these are only minor concerns compared to the magnet test, until you accidentally leave aluminum foil on an induction stove and it starts to melt.

It should be noted that cookware needs some inductive metal in it, but not all of it needs to be an inductive metal.  Some cookware designed for an inductive stove has an iron bottom with aluminum sides because aluminum conducts the heat better for more even cooking. Clad cookware with a mixture of the right metals works too.