Book excerpt: The current research interest in heavy fermion (HF) materials is for their unconventional superconductivity, their quantum critical behavior and the breakdown of Fermi liquid (FL) theory. Presently, we lack a universal understanding of the breakdown of the FL behavior in these materials. However, there are evidences which suggest that the breakdown of the FL behavior and the unconventional superconducting (SC) pairing could be the result of a zero temperature phase transition, taking place at a quantum critical point (QCP). Heavy fermions are f-electron materials in which local moments at each lattice site interact with the spin of the conduction electrons sitting at that site via an exchange coupling. There are two energy scales that result from this interaction, the Kondo temperature TK (temperature below which the local moments are screened by the spins of the conduction electrons), and the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction, which characterize the induced coupling between two local moments. These two energy scales can be tuned by external parameters such as magnetic field, pressure, and chemical substitution. Such a tuning provides an opportunity to study the rich physics of these materials. This dissertation work presents experimental and theoretical studies on ab unique member of Ce-115 family of heavy fermions, i.e., Ce1-xYbxCoIn5. In the Ce-115 family of HFs, Cerium (Ce) contributes the f-electrons to form a Kondo lattice. The substitution of Ce ions by other rare earths is a widely used approach to study this system. Our selection of ytterbium for substituting Ce-site is unique in the sense that Yb appears in the intermediate valence state in this system (unlike any other substitution), thus giving rise to many of the unusual properties, which helped in the understanding the underlying physics of SC pairing, quantum criticality and non-Fermi liquid behavior in this HF. For the purpose of the studies presented in this dissertation, we utilized electronic, magneto- and thermal transport measurements. These measurements were done under high magnetic fields and pressures wherever needed. We developed a new method to identify the field-induced QCP in this material by studying its normal state. We utilized this method to locate QCP in the parent compound CeCoIn5 and determined its evolution with Yb doping in Ce1-xYbxCoIn5. Our findings show that quantum criticality in this system is suppressed by doping with Yb and a zero field QCP is obtained for the x = 0.20 Yb-doping level. Our studies also show the evolution of the many-body electronic state as the Kondo lattice of Ce moments is transformed into an array of Ce impurities with Yb-doping. Specifically, we observe a crossover from the predominantly localized Ce moment regime to the predominantly itinerant Yb f-electronic state regime. In the crossover regime, the magneto-transport behavior of the system indicates single impurity behavior of Ce ions. This result is surprising because the resistivity and specific heat measurements suggest significant amount of coherent scattering in the system. We attribute this unusual behavior to the hybridization of conduction electrons with mixed valence Yb ions, giving rise to an intermediate energy scale (TK ~ 14 K) between the single impurity regime of Ce and Ce Kondo lattice regime. Even more intriguing are the results at even higher Yb-doping levels. Large enough Yb concentrations show an increased coherence, unlike any other member of the Ce-115 family. We also identified another QCP at a higher Yb concentration of x = 0.75. An equally interesting feature in the doping dependence of this compound is the survival of NFL behavior throughout the phase diagram. The sub-linear temperature dependence of resistivity across the whole range of Yb concentrations suggests the presence of an unconventional scattering mechanism for the conduction electrons. Thus although the quantum spin fluctuations are suppressed at around 20 % of Yb doping, the NFL behavior is observed for the whole family. Our finding of an additional high doping QCP very well explains the large value of the Sommerfeld's coefficient and the persistent NFL behavior over the whole Yb-doping range. Given the complete suppression of the antiferromagnetic fluctuations for x > 0.20 and the very robust coherence and superconductivity, the possible electron pairing mechanism may involve an exchange of virtual magnetic fluctuations or a more unconventional mechanism involving virtual fluctuations into higher lying Ce crystalline field multiplets. We analyze theoretically the dependence of the superconducting critical temperature and Kondo lattice coherence temperature on pressure for both cases of clean and disordered systems. We use the approach of the large-N mean field theory, which works very well for Kondo lattice systems.