Rational function interpolation?

Question

We know that $n+1$ points is enough to completely determine a polynomial of degree $n$. Are there any techniques which says that a certain number of points is enough to completely determine a rational function $r(x)=\frac{p}{q}$ where $p,q\in\mathbb{R}[x]$? What if we know a bound on the degree of the numerator or denominator?