Heavy fermion metals are intermetallic systems of certain rare-earth or actinide elements, whose partially-filled f electrons interact with each other strongly and form localized magnetic moments. The latter are coupled to weakly correlated conduction electrons by their spin, a process known as the Kondo coupling. These strongly correlated systems can be readily tuned nonthermally to produce quantum phase transitions, and often exotic phases emerge in the vicinity of quantum critical points (QCPs). An outstanding question is what happens when the strong correlations interplay with a large spin-orbit coupling.

In parallel, topological metals are a fascinating class of states, in which topologically protected and dissipationless transport makes them attractive for new electronic devices. In theoretical models of non-interacting electrons, such phases are favored by certain types of couplings, as well as by the particular set of symmetries for the geometric space that electrons move through (i.e. space group of a crystal lattice). Along with efforts to understand the required conditions for a topological phase, there has been an extensive search for weakly correlated materials platforms. What happens to topological metals in strongly correlated settings remains an outstanding open problem.

In this thesis, the heavy fermion systems are proposed to explore topological metallic phases driven by strong correlations. I theoretically demonstrated the existence of topological metallic phases by studying several types of Kondo lattice models. Importantly, the large Coulomb energy scale in these systems has several consequences for how topological metals behave.

In the first part of this thesis, I study the change in anomalous Hall conductivity (AHC) across a QCP in frustrated Kondo lattices. The frustration in the interactions between magnetic moments leads to time-reversal symmetry breaking (TRSB) chiral spin liquid phases, creating a highly singular Berry curvature field that influences the conduction electrons and their transport through the Kondo coupling. The QCP divides the Kondo screened phase with a large Fermi surface, from the Kondo destruction phase with a generically reconstructed Fermi surface. By studying this scenario on both the square lattice and kagomé lattice, I discovered that if the magnetic unit cell has an odd number of sites, the Fermi surface undergoes radical reconstruction in volume, resulting in a sharp jump of the AHC. The implications for the metallic pyrochlore heavy fermion iridate Pr2Ir2O7 are explored.

The second part of this thesis advances a correlation-driven topological metal, the Weyl-Kondo semimetal (WKSM). My theoretical work proceeded contemporaneously with experiments in a heavy fermion metal Ce3Bi4Pd3. I focused on a three-dimensional nonsymmorphic and noncentrosymmetric Kondo lattice. I show that the ground state is topologically trivial in the absence of the Kondo coupling, but is driven to be a topologically nontrivial Weyl semimetal by the Kondo effect. In the ensuing WKSM phase, a new ``Kondo-pinning'' effect fixes the Weyl nodes to the Fermi energy. Several distinguishing strong correlation effects are also shown for the WKSM. I then study the topological phases produced when a TRSB Zeeman field is introduced. The Weyl-Kondo nodes move and annihilate, leading to multiple phases. I study and discuss the relevance of my theoretical findings to Ce3Bi4Pd3, and propose experimental signatures which take advantage of the Kondo pinning mechanism.