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Away from the surface of the Earth or other planet; in opposite direction to the downward pull of gravity.

“I looked up and saw the airplane overhead.”

To or at a physically higher or more elevated position.

“All day we climbed up and up.”

To a higher level of some quantity or notional quantity, such as price, volume, pitch, happiness, etc.

“Cheer up, the weekend's almost here.”

To or in a position of equal advance or equality; not short of, back of, less advanced than, away from, etc.; usually followed by to or with.

“A stranger came up and asked me for directions.”

(intensifier) Used as an aspect marker to indicate a completed action or state; thoroughly, completely.

“Can you sum up your research?”

To or from one's possession or consideration.

“I picked up some milk on the way home.”

North.

“I live in Florida, but I'm going up to New York to visit my family this weekend.”

Towards or at a central place, or any place that is visualised as 'up' by virtue of local features or local convention, or arbitrarily, irrespective of direction or elevation change.

“He lives up by the railway station.”

Towards the principal terminus, towards milepost zero.

Aside, so as not to be in use.

“to lay up riches; put up your weapons”

Against the wind or current.

(Cartesian graph) In a positive vertical direction.

Relatively close to the batsman.

“The bowler pitched the ball up.”

(bartending) Without additional ice.

“Would you like that drink up or on ice?”

(academia) To university, especially to Cambridge or Oxford.

“She's going up to read Classics this September.”