High Voltage Resistors

Even such a mundane component as a resistor has special requirements when it comes to high voltage applications. The usual little 1/4 watt carbon film resistor used in most other electronics is only rated to 250 or 500 volts, a far cry from the kilovolt levels needed. The voltage limitation is usually set by power dissipation issues: a 10K resistor with 1 kV across it dissipates 100 watts! And, of course, the physical length of the device of around a centimeter means that around 5-10 kV, arcing around the resistor body is a significant problem. There are a number of manufacturers of resistors intended specifically for high voltage applications, and of course, you can construct a resistor suitable for your application.

Commercial products

High resistance, low power - These are typically used for measuring high voltages as part of a voltage divider (or as a meter series multiplier). Generally constructed by thin film techniques on a ceramic substrate of appropriate size for the voltage rating. Suppliers include Caddock, Gigohm, ....
High precision - Used in precision voltage dividers for measurement applications.
High power - Typified by the products of the Kanthal Globar (formerly Cesewid, formerly Carborundum), these units are designed to take large peak or average powers and are often constructed so they are non-inductive. A typical application would be as a current limiting device in a capacitor discharge circuit, components in pulse forming/shaping networks, or as a energy dump load. Suppliers include Globar and Maxwell Labs..

Home built

Using conventional products well beyond their catalog ratings - The power dissipation and breakdown voltage issues may be addressed by potting or immersing the resistor in some suitable substance: silicone, oil, high pressure gas, etc. You'll need to do some engineering and/or testing on the system to make sure it will work as planned, e.g. potting in silicone may increase the breakdown voltage, but will reduce the max power dissipation. There are some good reasons not to use wirewound power resistors, even though their nominal breakdown voltage is fairly high.
Strings of small low voltage resistors work very nicely, although their construction is labor intensive. This is a popular way to make high precision voltage dividers, because you can inexpensively obtain high precision resistors, and the whole assembly can be potted to reduce corona problems. Important design issues which need calculation include power dissipation, breakdown voltage, and overall precision.
Water resistors - Actually water and some ionic salt, these are usually self constructed and are generally of low resistance. Accurate resistance control and stability is hard to achieve, but there are many applications which don't require accuracy, including charging resistors for capacitor banks and energy dump loads. I've gathered some design criteria, equations and a discussion on construction details into a summary.