Across, Over or Through

Across and over can both be used to mean the other side of a line such as a river, road, bridge, etc.
Across is used to mean the other side of a flat area or surface.
Over is used to mean the other side of something high.
Through is used for movement in a three-dimensional (3D) space.

• His house is across/over the river.
• He walked across the desert.
• The thief climbed over the wall.
• They went for a walk through the forest.