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What is the right way to express sadness in Russian?
I often hear the expression мне грустно, which is formed with an adverb if I'm not mistaken.
Recently I also heard about the verb грустить and its first person singular я грущу, which in my opinion could also be used to express sadness. Is this form more often used in sentences and not in fixed expressions?
Then there's also a third form, я гру́стен. I've never really heard anyone using this expression, but theoretically it's a short form adjective and these are normally used for expressing current moods.

I appreciate all your answers, thank you for helping.

I would recommend you to explore the meanings of the words "печаль" and "тоска".
Е.g., "Обуяла меня печаль, одолела меня тоска."

Also of use are the verbs "печалиться" and "тосковать".

When the reason is known or implied, you can say "опечален".
Е.g., "опечален я Вашим поведением, сударыня".

It can also be said: "эти события повергают меня в тоску"

Another pertinent saying is: "тяжёлые мысли".
E.g., "Уже который день он ходил, погружённый в тяжёлые мысли."
(Оn this special occasion, "который" = "не первый".)

Besides, you can say: "тяжко на душе", "тяжело на душе", "муторно на душе"
(the latter saying being a bit more conversational).

While мне грустно neutrally expresses your state (I'm sad), я грущу means you are in a process of “being sad”.

As for me, я грущу sounds like a whim or an irony (like, “don't bother me now; I'm sadding for a short time, but then I'll be available for others”). So it should not be used.

The verb as opposed to the adverb seems more often than not to appear accompanied by an object appended with the prepositions по (which governs either Dative or Prepositional cases) or o/об (which governs Prepositional case) - грустить (п)о ком/чём-л. | по кому/чему-л.