  We present adaptive optics assisted integral field spectroscopy of 34
star-forming galaxies at $z$ = 0.8-3.3 selected from the HiZELS narrow-band
survey. We measure the kinematics of the ionised interstellar medium on $\sim$1
kpc scales, and show that the galaxies are turbulent, with a median ratio of
rotational to dispersion support of $v$/$\sigma$=0.82$\pm$0.13. We combine the
dynamics with high-resolution rest-frame optical imaging and extract emission
line rotation curves. We show that high-redshift star-forming galaxies follow a
similar power-law trend in specific angular momentum with stellar mass as that
of local late type galaxies. We exploit the high resolution of our data and
examine the radial distribution of angular momentum within each galaxy by
constructing total angular momentum profiles. Although the stellar mass of a
typical star-forming galaxy is expected to grow by a factor $\sim$8 in the
$\sim$5 Gyrs between $z$$\sim$3.3 and $z$$\sim$0.8, we show that the internal
distribution of angular momentum becomes less centrally concentrated in this
period i.e the angular momentum grows outwards. To interpret our observations,
we exploit the EAGLE simulation and trace the angular momentum evolution of
star forming galaxies from $z$$\sim$3 to $z$$\sim$0, identifying a similar
trend of decreasing angular momentum concentration. This change is attributed
to a combination of gas accretion in the outer disk, and feedback that
preferentially arises from the central regions of the galaxy. We discuss how
the combination of the growing bulge and angular momentum stabilises the disk
and gives rise to the Hubble sequence.
