  The explosion of ultra-stripped stars in close binaries can lead to ejecta
masses < 0.1 M_sun and may explain some of the recent discoveries of weak and
fast optical transients. In Tauris et al. (2013), it was demonstrated that
helium star companions to neutron stars (NSs) may experience mass transfer and
evolve into naked ~1.5 M_sun metal cores, barely above the Chandrasekhar mass
limit. Here we present a systematic investigation of the progenitor evolution
leading to ultra-stripped supernovae (SNe). In particular, we examine the
binary parameter space leading to electron-capture (EC SNe) and iron
core-collapse SNe (Fe CCSNe), respectively, and determine the amount of helium
ejected with applications to their observational classification as Type Ib or
Type Ic. We mainly evolve systems where the SN progenitors are helium star
donors of initial mass M_He = 2.5 - 3.5 M_sun in tight binaries with orbital
periods of P_orb = 0.06 - 2.0 days, and hosting an accreting NS, but we also
discuss the evolution of wider systems and of both more massive and lighter -
as well as single - helium stars. In some cases we are able to follow the
evolution until the onset of silicon burning, just a few days prior to the SN
explosion. We find that ultra-stripped SNe are possible for both EC SNe and Fe
CCSNe, and that the amount of helium ejected is correlated with P_orb - the
tightest systems even having donors being stripped down to envelopes of less
than 0.01 M_sun. We estimate the rise time of ultra-stripped SNe to be in the
range 12 hr - 8 days, and light curve decay times between 1 and 50 days.
Ultra-stripped SNe may produce NSs in the mass range 1.10 - 1.80 M_sun and are
highly relevant for LIGO/VIRGO since most (possibly all) merging double NS
systems have evolved through this phase. Finally, we discuss the low-velocity
kicks which might be imparted on these resulting NSs at birth. [Abridged]
