In this paper, we elaborate on the symmetric teleparallel gravity (STPG) written in a non-Riemannian space-time with nonzero nonmetricity, but zero torsion and zero curvature. First, 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 coincide with those of the Riemannian space-times. 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.