If you are purchasing a home with a mortgage, you will have to have an appraisal for the bank to determine if it will give you a loan provided you have jumped through all the other hoops. However, if you are a cash buyer, the answer to this question most of the time is "It depends."
If you want to make certain that you are paying a fair price for the home based on recent comps and an appraiser's professional opinion, then obviously you will want to get an appraisal. But what do you do when you find the absolute dream home, it has gone to multiple offer status and you are more than happy to pay the price you are offering? That is the time to step up to the plate and remove the appraisal contingency.
Many sellers, when offered cash, will automatically move the offer to the front of the line but when they see the appraisal contingency they relegate it back to the same yard-line as the offer with a mortgage contingency. If your offer is in the ballpark and you are cash with no appraisal contingency, the seller is often more interested in your deal because once the home passes inspection they know they are on the road to closing. The last two buyers I have worked with have been successful with the cash, no appraisal strategy and were not looking for a bargain or to "steal" the property. They found homes they loved, realized that there was not alot of inventory that appealed to them and they were willing to pay market value instead of haggling. Not every buyer has the luxury of paying cash or not worrying about an appraisal but when the right house comes along, it makes sense to achieve your dream and not settle for second best.