Shadouge

Shadouge is the het ship between Shadow the Hedgehog and Rouge the Bat from the Sonic the Hedgehog series.

Games
The two are teammates and current best friends in Sonic Adventure 2 and Sonic Heroes and often work together during mission stages, but are not seen together outside of duty alliance and do not spend time in one another's presence unless it is business. Nonetheless, Shadow is one of the few people Rouge cares about, other than herself.

Pre-Super Genesis Wave
Rouge has flirted with Shadow a number of times, and has displayed interest in him since he saved her live on Prison Island. Rouge has called Shadow 'handsome' from time-to-time. Like Knuckles, Shadow does not often accept her flirtation and other times, he simply brushes them off, but Shadow tends to smile around her often and does care for her well-being. Rouge also occasionally teases him by saying he mopes too much.

Post-Super Genesis Wave
They have demonstrated a strong connection with each other, transcending their status as teammates; Rouge is deeply concerned for his welfare. During the erased events of the timeline where they engaged Mephiles, she swore to always stand by him. This does not mean that she does not at times find him infuriating, though she is not above making jabs at him. Despite this, she does care deeply about him, and seeks to help him in whatever circumstances.

Sonic X
They didn't work well at first. However, during their partnership, Shadow and Rouge started to cooperate with each other better. Thus, Rouge is the closest person Shadow has, since Maria passed away. Proving that despite their occasional conflict, Shadow and Rouge geniunely respect and care for the well-being of one another.

Shadow is shown to care about Rouge, while she sometimes likes to tease Shadow for his moody attitude.

Fanon
Shadouge is a popular ship among the Sonic fanbase. Its greatest rivaling ships are Knuxouge for Rouge and Shadaria for Shadow.

This pairing is favored by some fans over Knuxouge, though other fans say that Shadow and Rouge don't seem to have a romantic interest in each other, but rather a brother-sister relationship. Others simply say that they don't have a brother-sister relationship or a close-friend respect, but a close ally relationship. Their interactions are interpreted differently by fans.

Tumblr

 * shadouge tag on Tumblr

AO3

 * Rouge the Bat/Shadow the Hedgehog tag on Archive of Our Own

DeviantArt

 * shadouge tag on DeviantArt
 * Shadouge-Lovers-Club fanclub on DeviantArt
 * SxA-SxLOVExR-SxB fanclub on DeviantArt