The question of how do the properties of nuclei evolve with increasing excitation energy and angular momentum is one of the current frontiers in nuclear physics. State of the art $gamma$-ray detector systems have been used to investigate this question, in a series of rare-earth nuclei with mass extit{A}$sim$158. Significant extensions to the high-spin excitation spectrum of the $N$=91, 92, 93 isotopes $^{157,158,159}$Dy have been achieved using the high-efficiency $gamma$-ray spectrometers EUROBALL and GAMMASPHERE. These nuclei were populated via weak 3$n$ or $alpha xn$ exit channels in fusion evaporation reactions. In $^{157}$Dy, the yrast band has been extended to extit{I}$^pi$=$frac{101}{2}^{+}$ (tentatively to $frac{105}{2}^{+}$) with four sideband structures (two of which are new) observed in the 35$-$50 $hbar$ spin range. In $^{158}$Dy, three bands have been extended to 42$^{+}$ (44$^{+}$), 40$^{-}$, and 41$^{-}$ (43$^{-}$), while in $^{159}$Dy the yrast band is observed to $frac{81}{2}^{+}$ ($frac{85}{2}^{+}$). A total of 84 (99) new transitions, including 2 new bands, were added to the level schemes of $^{157,158,159}$Dy. The high-spin behavior and band crossing systematics of the Dy isotopes and of the neighboring $N$=91, 92, and 93 isotones are discussed in terms of rotational alignments and shape transitions. Cranked Nilsson-Strutinsky calculations without pairing have been performed for detailed comparisons with the very high-spin states observed in $^{157}$Dy. Results on $^{157,158,159}$Dy have been published in extit{Phys. Rev. C.} Moreover, the angular-momentum induced transition from a deformed state of collective rotation to a non-collective configuration has been studied. In $^{158}$Er this transition manifests itself as favored band termination near extit{I}$approx$45$hbar$. The feeding of these band terminating states has been investigated for the first time using the GAMMASPHERE spectrometer. A large number of weakly populated states, lying at high excitation energy, that decay into these special states have been discovered. Cranked Nilsson-Strutinsky calculations suggest that these states arise from weakly collective configurations that break the $Z$=64 semi-magic core. Additionally, a new frontier of discrete-line $gamma$-ray spectroscopy at ultra-high spin has been opened in the rare-earth nucleus $^{158}$Er. Two rotational structures, displaying high moments of inertia, have been identified, which extend up to spin $sim$65$hbar$ and bypass the band-terminating states in these nuclei near extit{I}$sim$45$hbar$. Cranked Nilsson-Strutinsky calculations suggest that these structures arise from well-deformed triaxial configurations that lie in a valley of favored shell energy, which also includes the well-known triaxial strongly deformed bands in $^{161-167}$Lu. Overall, 182 (209) new transitions, including 10 new bands, were placed in the greatly augmented level scheme of $^{158}$Er, as a result of our work in this thesis. Four of the new bands are based on high$-$ extit{K} quasiparticle excitations, which provide a stringent test of the Cranked Shell Model. This enables the investigation and interpretation of many different quasiparticle configurations from their alignment properties and band crossings systematics. Results on $^{158}$Er have been published in extit{Phys. Rev. Lett.} and extit{Phys. Scr.} Finally, a local experiment, using the FSU tandem accelerator and the FSU $gamma$-ray detectors, was performed to investigate the odd-odd nucleus $^{158}$Tb. Unfortunately, no new significant results on the latter were obtained except for the tentative assignment of a new, strongly-coupled, rotational structure.}
