The subject of a sentence may be what parts of speech?

The subject is normally a noun or a pronoun.
However, it may also be the infinitive or the gerund form of a verb, or a clause.

"John was happy." (John is the subject).
"To travel to Europe is always fun." (To travel is the subject)
"Riding in his car made John happy." (Riding is the subject)

Interrogative and indefinite pronouns can represent subjects using a linking verb.
"What was his answer?" (subject is answer)
"There are no solutions." (subject is solutions)
"It was a dark and stormy night." (subject pronoun it)
"That we tried is what is important." (subject pronoun that)

Imperative sentences have an understood subject, which is you.
"Clean up the room" (you should clean up the room now)

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The subject of a sentence is the key and primary 'thing' that the writer/talker is writing/speaking about. In grammar, the word 'subject' refers to the person or thing that performs the action described by the verb.

For example, in the sentence "The cat sat on the mat", the verb is 'sat.'
So what thing performed the action of sitting? It was the cat.
So 'cat' is the subject in this sentence.

The complete subject of a sentence can be a noun phrase, meaning a phrase whose head word is a noun. The subjects in the following examples are enclosed between square brackets with the head noun italicized:

• [The new student] asked a question.
• [A question] was asked.
• [A student who studies linguistics] asked a question.
• Are [your students] enjoying the course?
• [He] thought that [the student] was very clever. (Here 'he' is the subject of the main clause and 'the student' is the subject of the embedded clause.)
• [I] expected [him] to go. (Here 'I' is the subject of the main clause and 'him' is the subject of the embedded clause 'him to go'.)

Various types of clauses can also function as the subject of a sentence when the main verb is a form of 'to be':

• [Playing cricket] is fun.
• [That I am alive] is obvious.
• [To leave now] would be a disaster.

Any kind of quoted text can also function as a subject when the verb is a form of 'to be':

• ["All work and no play"] was repeated on every page of the manuscript.
  

The subject of a sentence is a noun or pronoun.

It can be a pronoun or noun. :)

The subject is normally a noun phrase or a pronoun

noun - The dog ate the meat.
noun phrase - The two black dogs ate the meat.
pronoun - It ate the meat.
possessive pronoun - His dog ate the meat.