Given a family of algebraic varieties indexed by $t\neq 0$, one can ask for the ``limit variety'' at $t=0$. Grothendieck considered the case that the varieties are all subvarieties of a fixed projective space, and defined a limit {\it subscheme} which although very natural and useful is typically not a variety. In particular its geometry can be quite obscure. I'll give an alternate definition of the limit, which is still a variety but no longer ``sub''; we call it a {\bf branchvariety} of projective space, a branched cover of a subvariety. I'll give loads of examples. Grothendieck defined a moduli space of subschemes, the Hilbert scheme. I'll talk about the corresponding moduli stacks of branchvarieties, and the ways in which they're better behaved than Hilbert schemes. This work is joint with Valery Alexeev. {\bf The talk should be accessible to anyone who's seen e.g. the normalization of an algebraic variety; no scheme theory will be assumed.} (That's kind of the point.)