Two exotic modes of collective excitations of nuclei have been investigated in this work: the multiphonon excitations in the vibrational nucleus, 102Pd and the phenomenon of chirality in the 133Ce nucleus. The vibrational yrast states in 102Pd are described semiclassically as quadrupole running ('tidal') waves on the surface of the nucleus, and the propagating tidal wave interpreted as a rotating condensate of interacting, spin-aligned d bosons. The tidal wave concept has been investigated experimentally by measuring lifetimes of levels in the yrast band of the 102Pd nucleus using the Doppler shift attenuation method (DSAM). The extracted reduced transition probabilities, B(E2), for the yrast band display a monotonic increase with spin, in agreement with the interpretation based on rotation-induced condensation of aligned d-bosons, and the observed constant B(E2)/J ratios imply that the gain in angular momentum originates from the increase of the wave amplitude (deformation). In the second investigation, two distinct sets of chiral-doublet bands based on the three quasi-particle configurations π(1h11/2)2⊗ ν (1h11/2)-1 (higher-energy, negative parity) and π(1g7/2)-1(1h11/2)1 ⊗ν(1h11/2)-1 (lower-energy, positive parity) were identified in the nucleus 133Ce. The properties of these bands were observed to satisfy the established fingerprints of nuclear chirality and were found to agree with results of calculations based on a combination of the constrained triaxial relativistic mean field (RMF) theory and the particle-rotor model. They constitute a multiple chiral doublet (MχD), a phenomenon first predicted by RMF calculations. This study has provided the first experimental evidence for the existence of the MχD phenomenon, that represents, in general, a confirmation of triaxial shape coexistence.