Could cause more problems for normalisation (2NF, "Note that when a 1NF table has no composite candidate keys (candidate keys consisting of more than one attribute), the table is automatically in 2NF")
More unnecessary data duplication. If your composite key consists of 3 columns, you will need to create the same 3 columns in every table, where it is used as a foreign key.
Generally avoidable with the help of surrogate keys (read about their advantages and disadvantages)
I can imagine a good scenario for composite key -- in a table representing a N:N relation, like Students - Classes, and the key in the intermediate table will be (StudentID, ClassID). But if you need to store more information about each pair (like a history of all marks of a student in a class) then you'll probably introduce a surrogate key.