  We develop a construction of Engel stuctures on 4-manifolds based on
decompositions of manifolds into round handles. This allows us to show that all
parallelizable 4-manifolds admit an Engel structure. We also show that, given
two Engel manifolds M_1,M_2 satisfying a certain condition on the
characteristic foliation, there is an Engel structure on M_1#M_2#(S^2xS^2)
which is closely related to the original Engel structures.
