Mow the area you want to kill with your lawnmower. Place the lawn mower on its lowest setting to cut the grass as low to the ground as possible.
Wet the grass with a garden hose.
Place sheets of newspaper over the area you want to kill in an intersecting and overlapping pattern.
Spray the layer of newspaper with water and then add an additional layer of newspaper. In total, alternate between laying papers and spraying the paper until you have at least seven layers of newspaper. The newspaper will kill the grass and biodegrade, adding nutrients to the soil.
Place mulch or landscaping fabric over the newspaper. Mulch is great for a play area or a flowerbed. Landscaping fabric also works for these areas and is also great for rock beds.