Abstract The topological states of matter recognized as new phases of matter has become a central research field in condensed matter physics in the past two decades due to its exotic properties like nontrivial band topology, topologically protected edge and surface states, and topological phase transitions without symmetry breaking. In this thesis, we give a brief introduction to the topological Chern insulator and Weyl semimetal in Chapter 1, study the disorderinduced topological phase transitions in topological Chern insulators and Weyl semimetals in Chapters 2 and 3, and topological magnetic states in ferromagnetic materials in Chapters 4 and 5.   The disordered-induced topological phase transition was firstly found in topological Anderson insulator (TAI) suggests that when time-reversal symmetry (TRS) is maintained, the pertinent topological phase transition, marked by re-entrant $\frac{2e^2}{h}$ quantized conductance contributed by helical edge states, is driven by disorder. In Chapter 2, we show that when TRS is broken, the physics of TAI becomes even richer. The pattern of longitudinal conductance and nonequilibrium local current distribution displays novel TAI phases characterized by nonzero Chern numbers, indicating the occurrence of multiple chiral edge modes. Tuning either disorder or Fermi energy (in both topologically trivial and nontrivial phases), drives transitions between these distinct TAI phases, characterized by jumps of the quantized conductance from $0$ to $\frac{e^2}{h}$ and from $\frac{e^2}{h}$ to $\frac{2e^2}{h}$. An effective medium theory based on the Born approximation yields an accurate description of different TAI phases in parameter space.   In Chapter 3, the Quantum phase transitions of three-dimensional (3D) Weyl semimetals (WSMs) subject to uncorrelated on-site disorder are investigated through quantum conductance calculations and finite-size scaling of localization length. Contrary to a previous belief that a direct transition from a WSM to a diffusive metal (DM) occurs, an intermediate phase of Chern insulator (CI) between the two distinct metallic phases should exist due to internode scattering that is comparable to intranode scattering and persists at Weyl nodes for nonzero disorder. The critical exponent of localization length is $v\sim 1.3$ for both the WSM-CI and CI-DM transitions, in the same universality class of the 3D Gaussian unitary ensemble of Anderson localization transition. The CI phase was confirmed by quantized nonzero Hall conductance in the bulk insulating phase established by localization length calculations. The disorderinduced various plateau-plateau transitions in both WSM and CI phases were observed and explained by the self-consistent Born approximation. The initial studies of topological states were exclusively for electronic systems. It is now known that topological states can also exist for other particles. Indeed, topologically protected edge states have already been found for phonons and photons. In spite of active searching for topological states in many fields, the studies in magnetism are relatively rare although topological states are apparently important and useful in magnonics. In Chapter 4, we show that the pyrochlore ferromagnets with the Dzyaloshinskii-Moriya interaction are intrinsic magnonic Weyl semimetals. Similar to the electronic Weyl semimetals, the magnon bands in a magnonic Weyl semimetal are nontrivially crossing in pairs at special points (called Weyl nodes) in momentum space. The equal energy contour around the Weyl node energy is made up by the magnon arcs on sample surfaces due to the topologically protected surface states between each pair of Weyl nodes. Additional Weyl nodes and magnon arcs can be generated in lower energy magnon bands when an anisotropic exchange interaction is introduced. In Chapter 5, the chiral anomaly of Weyl magnons (WMs), featured by nontrivial band crossings at paired Weyl nodes (WNs) of opposite chirality, is investigated. It is shown that WMs can be realized in stacked honeycomb ferromagnets. Using the Aharonov-Casher effect that is about the interaction between magnetic moments and electric fields, the magnon motion in honeycomb layers can be quantized into magnonic Landau levels (MLLs). The zeroth MLL is chiral so that unidirectional WMs propagate in the perpendicular (to the layer) direction for a given WN under a magnetic field gradient from one WN to the other and change their chiralities, resulting in the magnonic chiral anomaly (MCA). A net magnon current carrying spin and heat through the zeroth MLL depends linearly on the magnetic field gradient and the electric field gradient in the ballistic transport.