  The present paper is devoted to the investigation of various properties of
transition fronts in nonlocal equations in heterogeneous media of ignition
type, whose existence has been established by the authors of the present paper
in a previous work. It is first shown that the transition front is continuously
differentiable in space with uniformly bounded and uniformly Lipschitz
continuous space partial derivative. This is the first time that regularity of
transition fronts in nonlocal equations is ever studied. It is then shown that
the transition front is uniformly steep. Finally, asymptotic stability, in the
sense of exponentially attracting front like initial data, of the transition
front is studied.

Analysis of PDEs;Classical Analysis and ODEs