  We define several notions of interpolation for vector bundles on curves and
discuss their relation to slope stability. The heart of the paper demonstrates
how to use degeneration arguments to prove interpolation. We use these ideas to
show that a general connected space curve of degree $d$ and genus $g$ satisfies
interpolation for $d \geq g+3$ unless $d = 5$ and $g = 2$. As a second
application, we show that a general elliptic curve of degree $d$ in
$\mathbb{P}^n$ satisfies weak interpolation when $d \geq 7$, $d \geq n+1$, and
the remainder of $2d$ modulo $n-1$ lies between $3$ and $n-2$ inclusive.
Finally, we prove that interpolation is equivalent to the---a priori
stricter---notion of strong interpolation. This is useful if we are interested
in incidence conditions given by higher dimensional linear spaces.
