Learn Russian: В городе ни следа прошлого. - In the city there is not a trace of the past.

The particle ни combined with the genitive case (следа) creates an emphatic verbless negation. It translates closer to 'not a single trace' or 'absolutely no trace.' If you used нет, as in В городе нет следа прошлого, it would be a standard factual statement meaning 'There is no trace.' Using ни makes it stronger and more expressive.

Why is следа in the genitive case?

In Russian, when you express the complete absence of something using the particle ни, the thing that is absent must take the genitive case. Here, the masculine noun след (trace) becomes следа.

How does this sentence work without a verb?

In the present tense, Russian routinely drops the verb 'to be' (быть). The construction ни + genitive is specifically designed to be used without a verb in the present tense to powerfully state that something doesn't exist. If you moved this to the past tense, a verb would appear: В городе не было ни следа прошлого.

What kind of word is прошлого and why is it also in the genitive case?

Прошлого is the genitive form of прошлое (the past). Прошлое is an adjective that has evolved to act as a noun (a substantivized adjective). It is in the genitive case here simply to link two nouns together, showing possession or association, meaning 'of the past' (a trace of the past).