We construct a general family of supersymmetric solutions in time-and space-dependent wave backgrounds in general supergravity theories describing single and intersecting p-branes embedded into time-dependent dilaton-gravity plane waves of an arbitrary (isotropic) profile, with the brane world-volume aligned parallel to the propagation direction of the wave. We discuss how many degrees of freedom we have in the solutions. We also propose that these solutions can be used to describe higher-dimensional time-dependent "black holes", and discuss their property briefly.The understanding of the fundamental nature and quantum properties of spacetime is one of the most important questions in theoretical physics. An example of such problems is the spacetime singularities that general relativity predicts especially in time-dependent setting. However, time-dependent solutions are rather difficult subject in the effective string theories [1]- [6]. Dilaton-gravity plane waves provide a rare example of tractable strongly curved (possibly singular) time-dependent space-time backgrounds. They also allow a formulation of (time-dependent) matrix theories of quantum gravity [7]- [36]. Other time-dependent brane solutions, though nonsupersymmetric, are discussed in [37,38].Brane supergravity solutions also plays important role [39]-[43], since they lead to the formulation of the AdS/CFT correspondence. Hence, it is important to derive supergravity p-branes embedded into dilaton-gravity plane waves.In a recent paper [44], the simplest of these solutions which are supersymmetric configurations corresponding to time-dependent extremal p-branes aligned along the propagation direction of the plane wave were obtained generalizing earlier work [6,29]. These solutions are restricted to single brane solutions. It has been known that this class of solutions can be extended to much more general intersecting brane configurations, and it would be interesting to see if the solutions can be generalized to such general configurations.In our previous paper [36], we have given closely related solutions for intersecting branes with wave, but we gave supersymmetric solutions without wave as examples of our solutions. This actually restrict possible solutions considerably. In this paper we revisit this class of solutions, and show that our previous solutions actually give a very general family of solutions which are wider than those for single brane in Ref. [44]. Our solutions involve many arbitrary functions and we examine how many degrees of freedom we have. We also discuss some physical applications of these solutions to black hole physics. In particular, we consider intersecting D1-D5 brane system, and show that we can effectively compactify it to five dimensions though close to the possible horizon six-dimensional nature of the solution reappears.This paper is organized as follows. In the next section, we briefly summarize our solutions derived in our previous paper [36], where we set wave profile to zero. This gave a strong restriction on the solu...