A quick recognition test: check whether the word sits before a noun or pronoun and connects it to the rest of the sentence. Take under the mat. The word under comes before the noun mat and links that noun phrase to the sentence, so under is a preposition.
It often marks place, time, movement, or reason.
A short word after a verb can be an adverbial particle with no object, as in give up meaning quit. The same word acts as a preposition when a noun object follows, as in give up smoking, where smoking is the object. Conjunctions such as before in before we leave link clauses instead of introducing a noun object.
Categories and Their Diagnostic Clues
Place. The word answers where? by tying a noun to a location. In the cat is under the table, under points to a spot and needs its noun to complete the picture.
Time. It answers when? and sits before a time word. In we met after lunch, after links the meeting to a specific noun, lunch.
Direction or movement. It shows a path toward or away from something. In she walked toward the station, toward takes the noun station as its destination.
Reason or purpose. It answers why? by attaching a noun to the action. In they stretched for flexibility, for connects the stretching to its purpose.
Manner or cause. Some grammarians group these differently, and one preposition can carry more than one meaning, so treat these labels as rough aids rather than fixed boxes. In they trembled with fear, with links the trembling to its cause.
Spatial relation between two things. Between is a useful check case because it typically needs two anchors. In the bakery sits between the bank and the café, both nouns follow it, confirming its preposition role.
The Stranded Word at the End
The test falters when the word sits at the end with nothing after it. What are you looking at? has no noun following at, yet at is still a preposition, not an adverb.
Refine the test: check whether the word connects to an object even when the object is moved or implied. In a question, the object appears earlier as what; in the papers you asked for, it is papers.
A quick repair helps. Turn the question back into ordinary order: you are looking at what. The noun phrase returns to its slot after at, confirming that at heads a prepositional phrase and takes that object.
Compare with a true adverb, where no object hides anywhere in the sentence. Please come in cannot be rearranged to attach a noun after in, so it stands alone as an adverb.
One caveat: grammars differ on how to label the stranded word, and some analyses describe the whole looking at unit differently. The practical test stays the same: if a moved or implied noun phrase fills the object slot, treat the word as a preposition.