  The main aim of this article is to show that a very general 3-dimensional del
Pezzo fibration of degree 1,2,3 is not stably rational except for a del Pezzo
fibration of degree 3 belonging to explicitly described 2 families. Higher
dimensional generalizations are also discussed and we prove that a very general
del Pezzo fibration of degree 1,2,3 defined over the projective space is not
stably rational provided that the anticanonical divisor is not ample.
