Hurwitz spaces (or Hurwitz schemes) parametrize covers of the projective line. One can do this in many ways. 


For example, one could fix the number $r$ of branch points, the degree $n$ of the cover and look only at simple covers of $\mathbf{P}^1$. This is usually denoted by $H\_{r,n}$. Fulton defined this space as a scheme over $\textrm{Spec} \mathbf{Z}$ and showed that $H\_{r,n} \otimes \mathbf{F}\_p$ is irreducible for $p$ big enough.


One could also fix a subset $B\subset \mathbf{P}^1$, the degree $n$ of the cover and look at covers of degree $n$ unramified outside $B\cup \{\lambda \}$, where $\lambda \in \mathbf{P}^1-B$ is allowed to vary. One can show that most curves arise as an irreducible component of such a space (Diaz, Donagi, Harbater).


One could also look at Galois covers with a fixed Galois group, etc.


In the end, there are many ways to parametrize covers of the projective line.


Are there any standard references that contain the basics of Hurwitz spaces?


At the moment I have at my disposal


Work of M. Romagny, J. Bertin and S. Wewers (available on Romagny's website). These are very *stacky*.


The article of Fulton *Hurwitz Schemes and Irreducibility of Moduli of Algebraic Curves*.


Notes by Brian Osserman available on his website *(The representation theory, geometry, and combinatorics of branched covers.)*.


The article of Diaz, Donagi and Harbater: *Every curve is a Hurwitz space*.


**Question.** What are the standard references for the basics of Hurwitz spaces/schemes?



Reference request: parametrizing covers of the projective line
There is really no universal reference, unfortunately, and what you should look at depends on what you're interested in. Are you interested in arithmetic or are you working over the complex numbers? Are you interested in compactification? Do you care mostly about simple branching or about more general branching or even about more general Galois groups than S\_n? Are you interested in these guys as subvarieties of M\_g, or as covers of M\_{0,n}, or both, or as abstract moduli spaces or....?


But I guess this is not an answer yet, so let me add to your already good list Harris and Morrison's book on algebraic curves, which has a nice discussion of admissible covers that should be very helpful for understanding how one kind of branched cover can degenerate to another. For the topologist's point of view, you might also look at any paper containing the words "braid group" and "Nielsen classes," or McReynolds's exposition of Thurston's proof of the congruence subgroup property for the braid group.


