From Teflpedia

Must is an English full modal verb that expresses obligation.[1]

Historically must was a preterite, but in Modern English, it's a base form with an uninflected preterite, so "must" is both the base form and its own preterite. But its behaviour is rather weird, as it has present simple and past perfect uses, but unlike other modal verbs tends not to be used in the present perfect or past simple.

In the present simple, must expresses obligation - "you must do your homework", and in the negative expresses prohibition, e.g. "you mustn't smoke in the kitchen". Though need to and have to are similar in meaning to must, "don't have to" and "don't need to" are different, as someone is not obliged.

In the past perfect, e.g. "they must have gone shopping" - it expresses deduction.

In the past simple, we tend not to use must but tend to use "had to"/"needed to" (positive) or "couldn't"/"weren't allowed to" (negative). e.g. *"When I was at school, we mustn't smoke", but "When I was at school we weren't allowed to smoke".

Must can also be used as a noun.

In the present perfect, is it comparable to will/may/can? Not sure.