  The arbitrariness of Yukawa couplings can be reduced by the imposition of
some flavor symmetries and/or by the realization of texture zeros. We review
neutrino Yukawa textures with zeros within the framework of the type-I seesaw
with three heavy right chiral neutrinos and in the basis where the latter and
the charged leptons are mass diagonal. An assumed non-vanishing mass of every
ultralight neutrino and the observed non-decoupling of any neutrino generation
allow a maximum of four zeros in the Yukawa coupling matrix $Y_\nu$ in family
space. There are seventy two such textures. We show that the requirement of an
exact $\mu\tau$ symmetry, coupled with the observational constraints, reduces
these seventy two allowed textures to only four corresponding to just two
different forms of the light neutrino mass matrix $M_{\nu A}/M_{\nu B}$,
resulting in an inverted/normal mass ordering. The effect of each of these on
measurable quantities can be described, apart from an overall factor of the
neutrino mass scale, in terms of two real parameters and a phase angle all of
which are within very constrained ranges. The masses and Majorana phases of
ultralight neutrinos are predicted within definite ranges with $3\sigma$
laboratory and cosmological observational inputs. The rate for $0\nu\beta\beta$
decay, though generally below the reach of planned experiments, could approach
it in some parameteric regions. Within the same framework, we also study Yukawa
textures with a fewer number of zeros, but with exact $\mu\tau$ symmetry. We
further formulate the detailed scheme of the explicit breaking of $\mu\tau$
symmetry in terms of three small parameters for allowed four zero textures. The
observed sizable mixing between the first and third generations of neutrinos is
shown to follow for a suitable choice of these symmetry breaking parameters.
