Whenever we think about the question of authorship in regard to books of the Bible, and in particular the Old Testament, we need to remember that there is often a wide number of different possibilities and critics are often sharply divided between different ideas.
However, with Song of Solomon, also called Song of Songs, it is widely believed that this book was written by King Solomon. This is of course strongly indicated in the first verse where it says:
The Song of songs, which is Solomon's.
If we have a look elsewhere in the Bible, and in particular in 1 Kings 4:32, we can see that King Solomon actually wrote many songs, and the way in which this one is refered to as "The Song of songs" in the opening verse seems to indicate that this was his best song out of all of them. It therefore seems logical to assume that we can believe this inference and take King Solomon to be the author of this richly passionate book of the Bible.