One-Dimensional (1D) nanostructures are generally defined as having at least one dimension between 1 and 100 nm. Investigations of their mechanical properties are important from both fundamental study and application point of view. Different methods such as in-situ tensile test and Atomic Force Microscopy (AFM) bending test have been used to explore the mechanical properties of 1D nanostructures. However, searching for reliable measurement of 1D nanostructures is still under way. In this dissertation, two methods, Atomic Force Acoustic Microscopy (AFAM)-based method and nanoindentation, were explored to realize reliable study of mechanical properties of two kinds of energy conversion-related nanomaterials: single crystalline rutile TiO₂ nanoribbons and alkaline earth metal hexaboride MB₆ (M=Ca, Sr, Ba) 1D nanostructures. The work principle of AFAM-based method is: while an AFM cantilever is in contact with a tested nanostructure, its contact resonance frequencies are different from its free resonance frequencies. The cantilever resonant frequency shift is correlated to the Young's modulus of the tested nanostructure based on Hertz contact mechanics. The measured modulus of BaB₆ nanostructures was 129 GPa, which is much lower than the value determined using the nanoindentation method. Due to the small load (120 nN) applied on the nanostructure during the experiment, the AFAM-based method may actually measure the mechanical property of the outside oxidation layers of BaB₆ nanostructures. Nanoindentation is capable of giving insights to both Young's modulus and hardness of bulk elastic-plastic materials. The assumptions behind this method are that the material being tested is a homogeneous half-space. Cares must be taken to extract properties of tested materials when those assumptions are broken down. Nanoindentation on a 1D nanostructure is one of such cases that those assumptions are invalid. However, this invalidity was not realized in most published work on nanoindentation of 1D nanostructures, resulting in unreliable data on mechanical properties of 1D nanostructures. In this work, factors which could affect measured nanostructure-on-substrate system modulus such as the selection of a substrate to support the nanostructure, the cross section of a nanostructure, the width-to-thickness ratio (or diameter) of a nanostructure, and the nanostructure-substrate contact mechanism were first subjected to a systematic experimental investigation. A Finite Element Modeling (FEM)-based data inverse analysis process was then proposed to extract the intrinsic modulus of nanostructures from measured system modulus. This data inverse process solved the intrinsic modulus of nanostructures by equalizing the simulated nanostructure-on-substrate modulus with the experimentally measured system modulus. In finite element simulation, another important aspect: the experimental indenter area function in addition to aforementioned other factors was carefully considered. Based on systematic experimental and numerical investigations, the Young's modulus of rutile TiO₂ nanoribbons, CaB₆ nanostructures, SrB6 nanostructures and BaB₆ nanostructures was determined to be 360, 175-365, 300-425 and 270-475 GPa, respectively. These numbers are the first reported mechanical properties for these nanomaterials. Besides the finite element simulation, an "analytical" solution to obtain a nanostructure-on-substrate system modulus is also presented. Compared to the finite element simulation, the solution could significantly reduce processing time for the data inverse method. It is applicable to a nanostructure with a width to thickness ratio larger than 4. This part of dissertation work clearly demonstrates that both experimental and numerical investigations are needed for studying of mechanical properties of 1D nanostructures by nanoindentation.