allerhand
all sorts
A large amount or variety of different things; many kinds. This word is not declined.
Auf ihrer Reise hat sie allerhand erlebt.
On her trip, she experienced all sorts of things.

all sorts / quite a lot
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all sorts
A large amount or variety of different things; many kinds. This word is not declined.
Auf ihrer Reise hat sie allerhand erlebt.
On her trip, she experienced all sorts of things.

The word is made up of 'aller' (from alle, all) and 'Hand' (hand). The original idea is a 'handful of everything,' which explains the meaning 'many different things' well.
Auf dem Markt gab es allerhand exotische Früchte zu kaufen.
At the market, there were all sorts of exotic fruits to buy.
'Allerhand' (all sorts / quite a lot) is an adjective with a special feature: it is never declined and always keeps its base form, no matter which noun it comes before or which case it is in.
Wir haben mit allerhand Problemen zu kämpfen.
We have to deal with all sorts of problems.
In the colloquial meaning 'unerhört' (outrageous), 'allerhand' often stands alone or with 'sein' (to be) as a short exclamation of surprise or outrage.
Zuerst kommt er zu spät und dann hat er die Unterlagen vergessen. Das ist wirklich allerhand!
First he arrives late, and then he forgot the documents. That is really quite something!
different / various
'verschieden' and 'allerhand' can both express that there are several different things. 'verschieden' sounds neutral and fits in almost any context. 'allerhand' emphasizes more strongly that there is a lot or a colorful mixture. It often sounds a bit more colloquial and lively. If you simply mean several kinds in a factual way, 'verschieden' is more natural. If you want to show that there are quite a lot of different things, 'allerhand' works well.

Mein Vater bringt von der Reise allerhand kleine Geschenke mit.
My father brings back all sorts of small gifts from the trip.
Für das Projekt brauchen wir verschiedene Unterlagen aus dem Büro.
For the project, we need various documents from the office.