Knaster received his Ph.D. degree from University of Warsaw in 1925, under the supervision of Stefan Mazurkiewicz. These two individuals are considered by many among the greatest point-set topologists of all time.
Knaster is famous for his discovery, in 1922, of the so-called hereditarily indecomposable continuum, later named the pseudo-arc, and the Knaster continuum, or buckethandle continuum.
In 1963 he received a Nagroda państwowa (national award). He was also known for his sense of humour. He organized meetings of mathematicians called knasteria which made history of this discipline in Poland.