(partly motivated by this question, but different: Degree of a Variety)

For a hyperelliptic curve $C$ of genus $g$ (over an algebraically closed field of characteristic not two) what is the smallest $d$ for which $C$ can be embedded in some $\mathbb{P}^n$ (I guess $n=3$ wlog) as a **smooth** curve of degree $d$? Does it depend only on $g$? Can anything be said for an arbitrary curve?