Reverse Osmosis (RO):
Water pressure is used to force water molecules through a membrane that has extremely tiny pores, leaving the larger contaminants behind. Purified water is collected from the "clean" side of the membrane, and water containing the concentrated contaminants is flushed down the drain from the "contaminated" side.  The average RO system is a unit consisting of a sediment/chlorine pre filter, the reverse-osmosis membrane, a water storage tank, and an activated-carbon post filter.  They cost from about $150 to over $1,500 for point of use systems.

The advantages of Reverse Osmosis include:
Reverse osmosis significantly reduces salt, most other inorganic material present in the water, and some organic compounds.  With a quality  carbon filter to remove any organic materials that get through the filter, the purity of the treated water approaches that produced by distillation.
Microscopic parasites (including viruses) are usually removed by properly functioning RO units, but any defect in the membrane would allow these organisms to flow undetected into the "filtered" water - they are not recommended for use on biologically unsafe water
Though slower than a carbon or sediment water filter, RO systems can typically purify more water per day than distillers and are less expensive to operate and maintain.
Reverse Osmosis systems also do not use electricity.  However, because they do require relatively high water pressure to operate, they may not work well in some emergency situations.

The disadvantages of Reverse Osmosis include:
Most point of Use RO units make only 12 - 24 gallons of treated water a day for drinking or cooking - which is ok for most homes since the treated water is stored in a tank for use.  
RO systems waste water. Two to four gallons of "waste" water are flushed down the drain for each gallon of filtered water produced.  
Some pesticides, solvents and other volatile organic chemicals (VOCs) are not completely removed by RO.  A good activated carbon post filter is recommended to reduce these contaminants.
Many conditions affect the RO membrane's efficiency in reducing the amount of contaminant in the water.  These include the contaminant concentration, chemical properties of the contaminants, the membrane type and condition, and operating conditions (like pH, water temperature, and water pressure). 
Although RO filters do not use electricity, they depend on a relatively high water pressure to force the water molecules through the membrane.  An electric booster pump can be used to increase water pressure if needed.  In an emergency situation where water pressure has been lost, these systems will not function.
* However, if a high quality activated carbon filter is used for the post filter, it could be disconnected and used to siphon water through in an emergency to reduce many contaminants.
RO systems require maintenance.  The pre and post filters and the reverse osmosis membranes must be changed according to the manufacturer's recommendation, and the storage tank must be cleaned periodically.
Damaged membranes are not easily detected, so it is hard to tell if the system is functioning normally and safely.

You might want to check  NSF International to see if the Reverse Osmosis system you are interested in purchasing is certified.

 A reverse-osmosis system is a good treatment option for people who have unacceptably high levels of dissolved inorganic contaminants in their drinking water which can not be removed effectively or economically by other methods. Water from shallow wells in agricultural areas that contains high nitrate levels is a good example of a situation where RO would make sense. Most people, however, who are using municipal water water that is subject to EPA regulations usually have acceptably low levels of nitrates. Maximum nitrate levels should be able to be determined by requesting a water quality report from your local water provider.

I have received a number of questions asking if distilled water is either bad for a person's health or beneficial for health relative to purified water that still contains ions (usually calcium and magnesium).  This question pertains equally well to water from a RO system., because most ions have been removed by the treatment. Click here to view my response to that question.