Across Vs Over

The prepositions 'Across' and 'Over' are used with similar meaning, but there are differences. 

Both Across and Over can be used to mean 'on or to the other side' of a line, river, road, and so on (position or movement related to things that are 'long and thin'. 

Examples: 

They walked across or over the road...

See if you can jump over or across the corridor...

Her room's just over or across the corridor...

They will be over/across the frontier by midnight...

Over is used for movements on or above water, but not in water. For example: How long would it take to swim across the river? (Not: ...over the river?)

Both Across and Over can mean 'on the other side of' a high barrier (like a hedge, a fence, a wall, a mountain range...), but only Over is used for a movement to the other side of something high. Compare the following: 

If we can be over/across the fence before sunrise, we have got a chance. 

When I last saw them, they were climbing very slowly over the fence. (Not: ...across the fence.)

Both Across and Over can be used for movement inside an area (Ex: fields, a desert, a dance floor). Note the following example: Who are those people wandering over/across the fields? 

However, we only use across when we mean 'from one side to the other of the area'. Note the following example: It took them five weeks to walk across the desert. (Not: ...over the desert.)

However, Over is not normally used for movement in a 3-D space (like a room). Note the following example: He walked across the room, smiling strangely. (Not: ...over the room.)

Note that the adverb 'over' has a wider meaning than the preposition 'over'. You can't say: Let's swim over the river to the church, but you can say: Let's swim over to the church...