  We report $^{75}$As nuclear magnetic resonance (NMR) / nuclear quadrupole
resonance (NQR) and transmission electron microscopy (TEM) studies on
LaFeAsO$_{1-x}$F$_{x}$. There are two superconducting domes in this material.
The first one appears at 0.03 $\leq$ $x$ $\leq$ 0.2 with $T_{\rm c}$$^{max}$ =
27 K, and the second one at 0.25 $\leq$ $x$ $\leq$ 0.75 with $T_{\rm
c}$$^{max}$ = 30 K. By NMR and TEM, we demonstrate that a $C4$-to-$C2$
structural phase transition (SPT) takes place above both domes, with the
transition temperature $T_{\rm s}$ varying strongly with $x$. In the first
dome, the SPT is followed by an antiferromagnetic (AF) transition, but neither
AF order nor low-energy spin fluctuations are found in the second dome. In
LaFeAsO$_{0.97}$F$_{0.03}$, we find that AF order and superconductivity coexist
microscopically via $^{75}$As nuclear spin-lattice relaxation rate (1/$T_1$)
measurements. In the coexisting region, 1/$T_1$ decreases at $T_{\rm c}$ but
becomes to be proportional to $T$ below 0.6$T_{\rm c}$, indicating gapless
excitations. Therefore, in contrast to the early reports, the obtained phase
diagram for $x \leq$ 0.2 is quite similar to the doped BaFe$_{2}$As$_{2}$
system. The electrical resistivity in the second dome can be fitted by $\rho =
{{\rho }_{0}}+A{{T}^{n}}$ with $n$ = 1 and a maximal coefficient $A$ at around
$x_{opt}$ = 0.5$\sim$0.55 where $T_{\rm s}$ extrapolates to zero and $T_{\rm
c}$ is the maximal, which suggest the importance of quantum critical
fluctuations associated with the SPT. We have constructed a complete phase
diagram of LaFeAsO$_{1-x}$F$_{x}$, which provides insight into the relationship
between SPT, antiferromagnetism and superconductivity.
