I will keep the answer short and to the point because I written a lot about the Savannah cat in different articles and you can see the main article by clicking on this link.
The Savannah cat can be illegal in some places because it is a wild cat hybrid. There is wild blood in this cat breed. The father, grandfather etc. is a serval, a medium-sized wild cat species. Some US states consider wild cat hybrids to be similar to the wild cat themselves and all states have laws governing the possession of exotic wild cat species. They can be seen as dangerous. Or if they escape the home they can breed with feral cats creating super-feral cats. That’s the fear from some authorities.
It should be said right away though that it is rather unusual for this cat breed to be illegal. Certainly fifth-generation Savannah cats are not illegal almost everywhere as I the recall. It is the high filial versions, the F1 and F2s which are large domestic cats containing a high percentage of wild cat genes. These are very exotic animals and states often have laws regarding the possession and ownership of exotic animals. But please check both at state and lower jurisdictions on the USA and in the UK and Europe the same advice applies.
As I recall, too, there are restrictive laws on ownership of hybrid cats in Hawaii, Massachusetts, Texas and Georgia. This applies to the Savannah cat. New York also has restrictions in place as I recall. Please check if you live in New York State or city as there may be updates. Here is a quote by me on the Savannah cat page:
“It is important to check the legality of owning a Savannah cat where you live. Most states in the US regard the Savannah cat as a domestic cat in following federal and United States Department of Agriculture advice. New York State doesn’t, although 5 generations from the wild serval are allowed in NY State but not NY city. Other states restrict ownership of hybrid cats (Alaska, Iowa, Hawaii, Massachusetts, and Georgia). Update Feb 2011: Georgia, Hawaii and Massachusetts have banned this cat in addition to New York).”
Conclusion: The Savannah cat is not automatically, by default illegal. It depends on the filial and the place. You should look up the law and/or make inquiries where you are just in case there are restrictions governing the ownership of a Savannah cat especially the high filial types. Don’t just dive in and assume they are authorized because you might be disappointed.