rapid
Allied, British, German, etc.

make
The regiment made an advance on the enemy lines.

order
The general ordered an advance to the front line.

halt, resist, stop

advance on
the advance on Madrid

advance to, advance towards/toward
the Russian advance towards/toward Berlin

big, considerable, dramatic, enormous, great, spectacular, substantial, tremendous
Hindu science made great advances in astronomy and mathematics.

important, major, notable, remarkable, significant
Recent years have seen significant advances in our understanding of the human genome.

rapid, steady
rapid advances in science and technology

recent
economic, educational, medical, political, scientific, social, technical, technological
The design incorporates the most recent technological advances.

theoretical

make
We need more money if we are to make any further advances in this area of science.

advance in
two major advances in medical science

advance on
an advance on the existing techniques

advance towards/toward
an advance towards/toward a better understanding of God

large
cash

give, pay
The publishers have paid me an advance.

get, receive

advance of
an advance of $10 000

advance on
He was paid £5 000 as an advance on royalties.

amorous (humorous), sexual

make
He made amorous advances to one of his students.

advance to

quickly, rapidly
cautiously, slowly

from
troops advancing from the south

into
The troops advanced into central Europe.

on, upon
The army advanced on the capital.

towards/toward
He advanced towards/toward me in aggressive style.

considerably, greatly, significantly
rapidly

beyond
Society needs to advance beyond prejudice and superstition.

Advance is used with these nouns as the subject:
army, rebel, tank, technology, troops

Advance is used with these nouns as the object:
agenda, argument, career, cause, concept, explanation, frontier, interest, knowledge, plot, profession, proposition, science, theory, thesis, understanding, view