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):

– 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.”

– 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 21. century.”
“Sie war der Herausforderung gewachsen.” – “She was ready for the challenge.”

Incidentally, the English word “challenge” comes to us from Latin (“calumnia” – “trickery”/“misrepresentation”) via Old French (“chalonge” – “slander”).