  We report on searches for neutrinos and antineutrinos from astrophysical
sources performed with the Borexino detector at the Laboratori Nazionali del
Gran Sasso in Italy. Electron antineutrinos ($\bar{\nu}_e$) are detected in an
organic liquid scintillator through the inverse $\beta$-decay reaction. In the
present work we set model-independent upper limits in the energy range 1.8-16.8
MeV on neutrino fluxes from unknown sources that improve our previous results,
on average, by a factor 2.5. Using the same data set, we first obtain
experimental constraints on the diffuse supernova $\bar{\nu}_e$ fluxes in the
previously unexplored region below 8 MeV. A search for $\bar{\nu}_e$ in the
solar neutrino flux is also presented: the presence of $\bar{\nu}_e$ would be a
manifestation of a non-zero anomalous magnetic moment of the neutrino, making
possible its conversion to antineutrinos in the strong magnetic field of the
Sun. We obtain a limit for a solar $\bar{\nu}_e$ flux of 384 cm$^{-2}$s$^{-1}$
(90% C.L.), assuming an undistorted solar $^{8}$B neutrinos energy spectrum,
that corresponds to a transition probability $p_{ \nu_e \rightarrow
\bar\nu_{e}}<$ 7.2$\times$10$^{-5}$ (90% C.L.) for E$_{\bar {\nu}_e}$ $>$ 1.8
MeV. At lower energies, by investigating the spectral shape of elastic
scattering events, we obtain a new limit on solar $^{7}$Be-$\nu_e$ conversion
into $\bar{\nu}_e$ of $p_{ \nu_e \rightarrow \bar \nu_{e}}<$ 0.14 (90% C.L.) at
0.862 keV. Last, we investigate solar flares as possible neutrino sources and
obtain the strongest up-to-date limits on the fluence of neutrinos of all
flavor neutrino below 3-7 ,MeV. Assuming the neutrino flux to be proportional
to the flare's intensity, we exclude an intense solar flare as the cause of the
observed excess of events in run 117 of the Cl-Ar Homestake experiment.
