The German word for “challenge” is Herausforderung (f.). The verb is herausfordern.
Literally this word translates to something like “to demand/require out (of something)” and is used in much the same way as the English “challenge” (except for the very modern use of alluding to physical handicaps).
Literal / Physical Examples
To start a fight (as in a duel):
Er hat mich zum Kampf herausgefordert. He challenged me to a fight.
Der Boxer war auf die Herausforderung nicht vorbereitet. The boxer was not prepared for the challenge.
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Figurative Examples
A task that requires/demands/asks a lot to be completed successfully:
Die Erderwärmung ist die größte Herausforderung des 21 Jahrhunderts. Global warming is the biggest challenge of the 21st century.
Sie war der Herausforderung gewachsen. She was ready for the challenge.
Fun Fact
Incidentally, the English word “challenge” comes to us from Latin (calumnia – trickery/misrepresentation) via Old French (chalonge – slanderd).