Lithium-Polymer (LiPo) batteries
Lithium polymer batteries are the primary reason the device can be as thin as it is, and run for as long as it can. However, they are also a constant source of worry because of what can happen if they are not properly handled.Safety
The first thing one should do when handling the batteries is to attach the batteries to a protection circuit. The protection circuits are relatively inexpensive (less than $2) and very small (ours were about 3 cm x 0.5 cm) so if you are not planning to arrange the batteries in an exotic fashion, it is much easier to just use one protection board per battery. For assembling multiple packs in series, there are several tips from the model airplane community and from people rebuilding laptop battery packs. One good source of information is "How to rebuild a Li-ion battery pack".
Charging
Lithium-Ion/Polymer batteries need a special charging chip to make sure the correct charging procedure is followed. It took me a while to understand the charging process, so I will attempt to be more clear here. The charging chip performs two roles, which are regulating the charging voltage the battery sees to 4.2V and limiting the charging current. If the voltage in the battery is low, the voltage difference between the battery and the external power source will be high, resulting in a high charge current. The charge chip will make sure that the current does not exceed the current the battery is rated for (generally the 1C current). Then, as the battery voltage rises as it charges, the smaller difference in voltage will cause the charge current to drop. The charge chip will observe the drop in charge current and shut down charging once the current drops to a low enough amount, in order to prevent overcharging.
Miscellaneous Advice
For prototyping purposes, it is generally a good idea to have a means to disconnect the battery from the system. The reason for this is that for debugging or for green-wire fixes, it is useful to have the system unpowered lest a short puts a current spike through part of the circuit.