Coins may be 
collected as objects in their own right for a number of reasons.  
The scientific study of coins is a branch of numismatics.
Some things that make coins special and therefore collectible are:
-  Engraving errors
-  Stamping errors
- Double strike
- Off metal
- Off center
- Clip
 
- Limited run
- Discontinued
- Foreign or World Coins
- Ancient
- Defunct nation
Collections are often built on coins within one of the above areas.  Other collections might consist of a single coin such as the U.S. Morgan silver dollar, with a coin from each mint for every date that Morgan dollars were minted.  
A type set contains one representative of each coin that was minted in a particular country.  A U.S. type set would include 2-cent, 3-cent pieces among the different kinds of 1-cent, 5-cent, etc. pieces.  Most people are not aware that the U.S. minted two and three cent pieces; fewer still have ever seen one.
 
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