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The identification of vibrational dynamics in the low-lying excitation spectra of deformed nuclei continues to be an open challenge in nuclear structure physics. In algebraic, phenomenological, or geometric models, the lowest vibrational shape affecting oscillations that can be built on a deformed ground state include the quadrupole ($\lambda = 2$) and the octupole ($\lambda = 3$) modes. Quadrupole oscillations are typically described in terms of $\beta$ and $\gamma$ one-phonon vibrational modes (lowest lying K$^{\pi}=0^+$ and K$^{\pi}=2^+$ bands), and excited bands (K$^{\pi}=0^+,2^+,4^+$ bands) as $\beta\beta$, $\beta\gamma$, and $\gamma\gamma$ two-phonon excitations, that is (2$^+ \otimes 2^+$) quadrupole phonons. The quadrupole mode of oscillations along the symmetry axis result in $\beta$ vibrations with K$^{\pi}=0^+$ and those breaking axial symmetry, result in $\gamma$ vibrations with a projection of K$^{\pi}=2^+$ on the symmetry axis. The discussions on the nature of numerous low-lying bands observed in the spectra of deformed nuclei continue. Are they indeed vibrations built on the ground state? Or are they coexisting minima of other shapes? The discussions have led to a reexamination of the nature of vibrational excitations. Vibrational excitations built on the ground state of a deformed nucleus are expected to show the same degree of deformation, the same dynamic and kinetic moments of inertia with some dispersion, and where there is enough information the same intrinsic quadrupole moment for transitions from the same band connecting to the ground state band.
There are a number of two-phonon vibrational excitations identified in deformed nuclei, two-phonon $\beta\beta$ vibrational type in $^{178}$Hf, and a potential $\beta\gamma$ type in $^{162}$Dy, and several cases of potential $\gamma\gamma$, K$^{\pi}=4^+$ bands in the rare-earth region of well-deformed nuclei. The ratios of the two-phonon to one phonon vibrational excitations energies are theoretically expected to be at twice the single phonon values. The B(E2) ratios of the two-phonon $\gamma\gamma$ to single-phonon to ground state values are expected to be 2.78 for K$^{\pi}=4^+$ and 5.0 for K$^{\pi}=0^+$. Results from current measurements and interpretations will be presented and discussed.