We study various aspects of D-branes in the two families of closed N=2
strings denoted by \alpha and \beta in hep-th/0211147. We consider two types of
N=2 boundary conditions, A-type and B-type. We analyse the D-branes geometry.
We compute open and closed string scattering amplitudes in the presence of the
D-branes and discuss the results. We find that, except the space filling
D-branes, the B-type D-branes decouple from the bulk. The A-type D-branes
exhibit inconsistency. We construct the D-branes effective worldvolume
theories. They are given by a dimensional reduction of self-dual Yang-Mills
theory in four dimensions. We construct the D-branes gravity backgrounds.
Finally, we discuss possible N=2 open/closed string dualities.Comment: 25 pages, Latex2