You can put arbitrary type-level restrictions on any argument in a function's signature. You can also interleave runtime with compile time to restrict things based on value. Julia's type system is fully parametric and allows values in the parameters.

If I write

    f(x :: Number) = x + 1
    f(x :: String) = x * x
    f(x :: Int)    = x - 1

this defines 3 separate methods for the function f. The Int method is actually more specific than the Number method because Int <: Number, so we get the following behaviour:

    julia> f(1.0)
    2.0

    julia> f("hi ")
    "hi hi "

    julia> f(2)
    1

Regarding this part of your question

> when you're looking at the signature, does it ensure that the operations performed on values inside are going to work?

I guess the answer depends on what you mean. Julia's interfaces are somewhat implicit, and we do not do ahead of time checks by default, so if I define my own type that is a subtype of Number, but that type does not have methods for addition, then f will error on that type.

However, the static checking package JET.jl will easily detect that this will happen statically at compile time.