Hilbert schemes of conics and extensions of hyperplane sections

After giving a rough idea of the main technical tools needed, I will construct a couple of Hilbert schemes of curves in $P^3$ [Harris]. I will then interpret such schemes as parameter spaces of extensions of lower dimensional objects (points) and make a few remarks. If time permits, I will give a brief overview of the problem of extending hyperplane sections [e.g. Reid].
The style of the talk will of course be very informal and I will try to explain in elementary terms most of the notions I will talk about.