Newton's method for solving nonlinear equations is known to converge quadratically when the starting guess is "sufficiently close" to the solution.
What is "sufficiently close"?
Is there literature about the structure of this basin of attraction?
For a single rational equation in the complex domain, the basin of attraction is fractal, the compelement of a so-called Julia set. http://en.wikipedia.org/wiki/Julia_set . For theory with some nice online figures, see, e.g.,
Even the ''globalized'' damped Newton method for $x^3-1=0$ has a fractal basin of attraction; see http://www.jstor.org/stable/10.2307/2653002 .
Thus there is little point in specifying in detail what is "sufficiently close" to the solution. If one knows bounds on the second derivatives, there is the Newton--Kantorovich theorem which gives lower bounds on the radius of a ball in which Newton's method converges, but except in 1D, these tend to be quite pessimistic.
Computationally useful bounds can be obtained using interval arithmetic; see, e.g., my paper
"Sufficiently close" is difficult to characterize considering that it gives rise to a class of fractals. Newton methods with globalization strategies such as line search and trust region extend the basin of attraction. If additional problem structure is available, such as in optimization, the assumptions necessary for convergence can be further weakened.