In English the phrase "all kinds of" is usually assumed to be an informal, hyperbolic way of saying "a large variety of", for example if I say "There are all kinds of animals in these woods," then you probably wouldn't think I mean that you can find sea slugs and penguins there. I'm wondering how this compares with "alle Arten von" and whether there is another German phrase that would be a better translation. Would it would be more accurate to drop the hyperbole and just say "viele Arten von"?
You could also translate "There are all kinds of animals in these woods" as such:
Es gibt allerlei Tiere in diesen Wäldern.
Es gibt allerhand Tiere in diesen Wäldern.
While they look very similar, the second one has an emphasis that there are actually a lot of animals in the woods. The first one doesn't.
Both sentences would be understood as "all kinds of animal species", even without the "species" clarification.
As a native speaker, I would use those sentences for writing. For talking, I think it's more common to use "alle möglichen Tiere" like Kilian suggested because it sounds a bit less formal, though all 3 variations are perfectly okay.
Es gibt viele Arten von Tieren in diesen Wäldern.
(from rastafile) is also good, but as he stated it's not very elegant. "Es gibt viele Tierarten in diesen Wäldern." is better.
You should avoid "Arten von" in biological context (=species), unless you mean it. "Seltene/geschützte Arten" in plant or animal context is "rare/protected species". So in German "Viele Arten" often is unlucky, because it collides; there is "Artenvielfalt" and "artgerechte Tierhaltung", nowadays.
But try allerart:
There are all kinds of animals in these woods.
Es gibt allerart Tiere in diesen Wäldern.
Duden.de says it is "veraltend" - true, that is why I hesitate a bit.
I still think it fits nicely in this case, because suddenly the biological collision is an advantage. You can not use "Alle Arten von", so you fall back to old "allerart".
The English original could be meant more in the sense of "besides sea-slugs, you will find just about anything in these forests". With "alle möglichen" and "allerlei" it tends in this direction.
God (or: Nature) created all kinds of animals
It is not really clear what kind of "all kinds of" is, but that is here exactly the point, in Genesis as in "these woods".
According to Tristram Shandy, when you open a drawer and find more than you were expecting, you do a typical "hmph!". With allerart Tiere I try to avoid that impression a bit.
Es gibt viele Tierarten in diesen Wäldern.
This is correct, but boring (the hyperbole is dropped)
Es gibt alle Arten von Tieren in diesen Wäldern.
correct, maybe, but naive.
Es gibt mancherlei Tiere in diesen Wäldern.
I was fearing duden's "veraltend" again, but now the first definition is:
verschiedene einzelne [ins Gewicht fallende] Dinge, Arten o. Ä. umfassend
ins Gewicht fallende: that is the point. Allerart Tiere, allerlei Pflanzen, allerlei Sportarten.
The yellow-winged darter is a dragonfly found ...
Die Hufeisen-Azurjungfer ist eine Libellenart...
(from wikipedia; I deleted the latin names which define the species)
This shows the subtle difference. Google translates "Libellenart" with "type of dragonfly". I find better "kind of dragonfly". In this sentence, German is easier, auf eine Art (=in some way ;)