  In this paper we elaborate on the symmetric teleparallel gravity (STPG)
written in a non-Riemannian spacetime with nonzero nonmetricity, but zero
torsion and zero curvature. Firstly we give a prescription for obtaining the
nonmetricity from the metric in a peculiar gauge. Then we state that under a
novel prescription of parallel transportation of a tangent vector in this
non-Riemannian geometry the autoparallel curves coincides with those of the
Riemannian spacetimes. Subsequently we represent the symmetric teleparallel
theory of gravity by the most general quadratic and parity conserving
lagrangian with lagrange multipliers for vanishing torsion and curvature. We
show that our lagrangian is equivalent to the Einstein-Hilbert lagrangian for
certain values of coupling coefficients. Thus we arrive at calculating the
field equations via independent variations. Then we obtain in turn conformal,
spherically symmetric static, cosmological and pp-wave solutions exactly.
Finally we discuss a minimal coupling of a spin-1/2 field to STPG.
