ACROSS and OVER are both prepositions and adverbs. They are in most cases interchangeable.
Look at these sentences:
They had to go across the river to get to their house.
We walked over the bridge in the misty morning.
However, when the meaning is ‘from side to side’, ACROSS is preferred:
I ran across the street.
Juliana folded her arms across her chest.
When moving from one side to another in a surrounding environment, across is replaced by THROUGH:
I made my way through the bushes.
The Red Riding Hood went through the woods to get to her grandma’s house.
Don’t use THROUGH when talking about periods of time. In these cases, OVER is preferred.
I haven’t seen Tom much over the last three years.