‘across’ ‘over’ and ‘through’ (as prepositions)

Across and over can both mean ‘on or to the other side of a line, river, road, bridge etc’.

You can use both in this way:
His village is just across the border.
He ran across the finishing line and raised his arms in victory.

But here’s the difference that you need to keep in mind:

Over is used to mean on or to the other side of something high.
Example:
The cat jumped over the fence and disappeared.

Across is used to mean movement from one side to the other side of a flat area or surface.
Example:
The river is too wide. We can’t swim across.

Use through for a movement in a three-dimensional space, with things on all sides.
Example:
We drove through the town in search of an antique shop.
BUT
We drove across the desert.

Do you see the difference?