Gentle slopes are usually okay except for very heavy rains. For steeper slopes where you have installed fabric and had a problem keeping the mulch in place, try this: Remove the mulch but leave the fabric in place.
Purchase netting such as Deer-X or Bird-X. Put 1/3 of the mulch back on the fabric to raise the netting slightly. Then put the netting over the mulch and use Gardeneer Garden Staples to hold it down. Put the remaining 2/3 of the mulch on top of the net. The net should hold the mulch in all except the heaviest of rains. The black net will not show, even if it is not completely covered.