To replace uses where you would use Matlab or R probably. I prefer Julia over Matlab or R. So data science. For production code however, it's not great since it has no static typing. Imagine having your production code crash mid-execution with the error "Function foo not found". Only dynamic languages can do that to you.

I broadly agree that it can be hard to nail down Julia's behaviour but it does have static typing and I think it is more subtle. Function arguments and variables can be concrete types e.g. if you were implementing an approximation for sin, you could restrict arguments to Float32 if you knew it was only suitably accurate for that type.