Using shadow puppets is a very old storytelling tradition around the world.

(What is the part of speech of 'storytelling'? Is it an adjective or a noun?)

In this sentence, "storytelling" is an adjective describing "tradition."

In this sentence, it's an adjective. (a participle)

To determine the part of speech of a word, you can consult a dictionary or look at how the word is used in a sentence. In this case, "storytelling" is being used as a noun because it is referring to an activity or practice. It is the act of telling stories.