The solar filament is one of the most important structures that lead to the destabilization of the solar corona, thereby driving the space weather in the Earth space. The dynamics of solar eruptive filaments is crucial for us to understand the physics governing the initiation of coronal mass ejections (CMEs). In this thesis we concentrate on kinking filaments and asymmetric eruptive filaments, which feature unique dynamic evolutions with implication of distinct initiation mechanisms. Kinking filaments with their warped axes are generally regarded as the 'fingerprint' of the MHD helical kink instability. Theoretical/numerical modelings of the kink instability in the solar context have raised a number of interesting issues which can only be fully addressed with detailed observational inputs. Our study on the kink evolution in a number of filament eruptions with a wide range of different natures provide a complete picture of how the kink instability works in the interactions of the filament with its magnetic environment. Our work has shown evidence supporting the writhing motion of the filament spine as a precursor of eruptive phenomena in the solar corona, and as a key component in regulating the nature of the eruption, in terms of full, partial or failed eruptions. The dynamic evolution of both kinking and asymmetric eruptive filaments has significant impacts on the production of hard X-ray emission. We have identified two types of asymmetric eruptive filaments: whipping-like and zipping-like, which are associated with the shifting of hard X-ray sources in different ways. Both can be understood in terms of how the highly sheared filament channel field responds to an external asymmetric magnetic confinement. In kinking filaments, our study suggests that two distinct processes take place during the kink evolution, leading to two types of HXR emission with different morphological connections to the overall magnetic configuration. Self-consistent, qualitative models are constructed in both studies. These results improve our understanding of the physical processes leading to the destabilization and eruption of solar filaments, and have significant impact on the modeling of the CME initiation and evolution.