big, considerable, enormous, great, huge, monumental, radical, real, serious, significant
daunting, difficult, formidable, stiff, tough
The army faced the daunting challenge of fighting a war on two fronts.

fundamental, key, main, major, primary, ultimate
fresh, new, unprecedented
constant, continuing, ongoing
future
exciting, interesting
special, unique
economic, engineering, environmental, intellectual, logistical, physical, political, technical, technological
Getting the instrument built and tested was a major technical challenge.

global

be, constitute, remain, represent
The competitive market represents significant challenges for farmers.

create, offer, pose, present, provide, raise
address, confront, face, handle, meet, respond to, rise to, tackle, take on, take up
The gallery has risen to the challenge of exhibiting the works of young artists.
He has taken on some exciting new challenges with this job.

overcome, solve
enjoy, relish, welcome

confront sb, face sb
the challenges facing nurses in casualty

arise, exist, remain
She could always be counted on when a challenge arose.

lie
The challenge lies in creating a demand for the product.

effective, serious, strong
direct
constitutional, legal
leadership, title
his title challenge to the heavyweight champion

be, constitute, present, represent
The demonstration represents a direct challenge to the new law.

file (esp. AmE), issue, mount, raise
plans to mount a leadership challenge within the party

accept, take up
I accepted his challenge to a game of chess.

answer
He was answering challenges to the government's policy.

beat off, fight off
Our team will have to fight off the challenge from better trained teams.

survive, withstand

challenge from
a challenge from the other political party

challenge to
a legal challenge to the President's power

directly, openly, publicly
The newspaper was directly challenging the government's legitimacy.

seriously, vigorously
effectively
She was effectively challenging the whole basis on which society was run.

successfully
The story was completely untrue and was successfully challenged in court.

constantly, continually, repeatedly

seek
Harley sought to challenge the jurisdiction of the court.

on
She challenged him on his old-fashioned views.

seriously
No one has seriously challenged the champion.

successfully, unsuccessfully

for
She was poised to challenge for the party leadership.

to
The count challenged him to a duel.

Challenge is used with these nouns as the subject:
case, movement

Challenge is used with these nouns as the object:
assertion, assumption, authority, belief, claim, competence, conception, convention, credibility, decision, detention, dogma, dominance, establishment, finding, hierarchy, idea, ideology, interpretation, legality, limit, logic, monopoly, myth, norm, notion, orthodoxy, perception, practice, preconception, prejudice, prerogative, racism, regime, stereotype, supremacy, theory, thesis, tradition, validity, view, will, wisdom