Symbols can be used to exploit the most unconscious-level of human desire, and they are able to create associations between a company and what the company stands for. Some might argue that the logo is in fact a symbol. And in some cases it is true. The symbol only logos are so famous that they don't need the company name attached to be recognizable.
“A picture tells a thousand words” is the main advantage of a symbol only logo, but it is probably not a good idea to use it for a brand when first launched. A good design along with the power of repetition can make a big impact, but it needs to be backed up by a company that already has a high level of brand recognition.
A symbol logo works well when you have a unique symbol in your industry and a global presence that can induce a universal, graphic symbol that speaks to you and audiences. Beware that a symbol can have meanings on many levels, and can also have different meanings in different cultures.
I've selected ten well-known logos that represent companies and organizations that are recognizable by their symbol alone. Do you recognize them all?