  We calculate the Josephson current between two one-dimensional (1D) nanowires
oriented along $x$ with proximity induced $s$-wave superconducting pairing and
separated by a narrow dielectric barrier in the presence of both Rashba
spin-orbit interaction (SOI) characterized by strength $\alpha$ and Zeeman
fields ($h$ along $\hat z$ and ${\bf B}$ in the $x-y$ plane). We formulate a
general method for computing the Andreev bound states energy which allows us to
obtain analytical expressions for the energy of these states in several
asymptotic cases. We find that in the absence of the magnetic fields the energy
gap between the Andreev bound states decreases with increasing Rashba SOI
constant leading eventually to touching of the levels. In the absence of Rashba
SOI, the Andreev bound states depend on the magnetic fields and display
oscillatory behavior with orientational angle of B leading to magneto-Josephson
effect. We also present analytic expressions for the dc Josephson current
charting out their dependence on ${\bf B}$, $h$, and $\alpha$. We demonstrate
the existence of finite spin-Josephson current in these junctions in the
presence of external magnetic fields and provide analytic expressions for its
dependence on $\alpha$, $\bf B$ and $h$. Finally, we study the AC Josephson
effect in the presence of the SOI (for $|{\bf B}|=h=0$) and an external
radiation and show that the width of the resulting Shapiro steps in such a
system can be tuned by varying $\alpha$. We discuss experiments which can test
our theoretical results.
