The majority of interactions in solids strongly depend on the interatomic distances. The application of pressure changes the lattice parameters and modifies the electronic and the phononic energy spectra of a material avoiding some of the undesirable effects induced by chemical doping, like lattice disorder, impurity phases and phase separations. Layered materials are particularly affected by the application of pressure because their anisotropic structure eases the contraction of the lattice along specific crystallographic directions. In this thesis I study the effect of pressure on the transport properties of two new layered materials: L4Fe2As2Te(1-x)O4 (L = Pr, Sm, Gd) superconductors and beta-Bi4I4 topological insulator. The discovery of Fe-based superconductors (Fe-SCs) provided a new material base for studying the mechanism of high temperature superconductivity. These materials present unexpectedly high critical temperatures (Tc ) despite the presence of magnetic atoms. The parent compounds of Fe-SCs are semimetals with an antiferromagnetic ground state in which the itinerant electrons form a periodic modulation of spin density called spin density wave (SDW). By changing the structural properties or by adding/removing carriers the SDW order can be suppressed and superconductivity emerges. Our group recently succeeded in synthesizing a new family of Fe-SCs that presents Tc up to 45 K upon optimum chemical doping and substitution. I performed a systematic study of the upper critical field and the pressure-dependent electrical resistivity of single crystals of Pr4Fe2As2Te(1-x)O4 and Sm4Fe2As2Te(1-x)O(4-y)Fy . Hall coefficient and magnetoresistance of Pr4Fe2As2Te(1-x)O4 revealed electrons as the dominant type of charge carriers. The results can be successfully fitted by a two-band model that allows the estimation of the temperature dependent electron and hole densities and mobilities. In view of the exceptionally high structural anisotropy of this new class of Fe-SCs I investigated the electronic anisotropy of Sm4Fe2As2Te(1-x)O(4-y)Fy by means of microfabrication techniques. The results show a ratio between the out-of-plane and in-plane electrical resistivities that is the highest reported so far in Fe-SCs. Topological insulators (TIs) form another class of materials that is currently attracting a lot of attention both for fundamental physics and potential applications. In TIs metallic surface states coexist with an insulating bulk. Most of the TIs known so far are either three-dimensional strongly bonded bulk materials or layered van der Waals crystals. Beta-Bi4I4 is a material composed of quasi-one-dimensional molecular chains that was theoretically predicted, and only recently experimentally verified, to be a novel strong topological insulator. Due to its particular structure, beta-Bi4I4 offers the possibility to study the interplay between different ground states like topological insulating phase, charge-density-wave and superconductivity. Here I present, for the first time, a study of the thermoelectric properties of beta-Bi4I4 single crystals. Electrical resistivity, Seebeck coefficient, thermal conductivity and Hall coefficient measurements reveal a possible charge density-wave-order (CDW) at low temperature. According to resistivity and thermoelectric power measurements, pressure suppresses the CDW order. Resistivity measurements performed in a diamond anvil cell up to 20 GPa demonstrate that superconductivity is induced in beta-Bi4I4 for pressure above 10 GPa. Preliminary X-ray diffraction measurements by synchrotron radiation, performed on the sample recovered at ambient pressure after the experiment, reveal that a new Bi-I phase is formed at high pressure. The possibility that the topological insulator state could coexist with superconductivity in beta-Bi4I4 is certainly a fascinating option but it remains for now an open question.