An auxiliary (classical/literary) attached to the imperfective form of verbs/adjectives to indicate conjecture or supposition; mainly found in literary or archaic Japanese.
その知らせは本当であろう（文語では「あらう」）と私は考えた。
I thought that the news was probably true (in literary form this would be 'arā').
古い日記には「今日は雨ならう」と書かれていた（古い書き方）。
An old diary had the phrase 'it will probably rain today' (old spelling/usage).
An auxiliary expressing will/intention; a marker of the speaker's volition. Mostly literary or replaced by modern equivalents in everyday speech.
ここで休まず、先へ進もう（口語では「進もう」＝意志・勧誘）／古い表記では「進まう」となることがある。
Without resting here, let's move on (colloquial 'susumō' = intention/invitation); in old orthography it may appear as 'susumau'.
An auxiliary used to express invitation. In modern Japanese this function is usually carried by forms like '〜よう' or '〜ましょう' rather than a bare '〜う'.
昔の戯曲では仲間を誘う台詞に「行かう」といった形が見られる（古い表現）。
In old plays you can see lines inviting comrades written like 'ikau' (an old form meaning 'let's go').