Systems, in which low energy excitations around the Fermi energy can be described by linear (Dirac) dispersion, are called Dirac materials. 3D topological insulators possess topologically protected conductive surface (2D) states with linear dispersion. In Dirac, Weyl and nodal line semimetals (topological semimetals), the valence and conduction bands touch each other in the symmetry or topologically protected touching points (or lines). Around this points (or lines) the bands are linearly dispersed. Accordingly, topological insulators and topological semimetals are Dirac materials that can work as a platform for studying, fundamentally very interesting, physics of Dirac fermions. Within this work the synthesis of high quality monocrystalline samples of 3D topological insulator Bi1.1Sb0.9Te2S Dirac semimetals Pb0.83Sn0.17Se and Cd3As2 and nodal line semimetals ZrSiS and HfSiS has been improved. The characterization and investigation of the Dirac fermion physics in samples were done by transport and magnetic measurements. Within this section the basic theory of topological insulators and topological semimetals is given. Surface states in 3D topological insulators are consequence of the nontrivial topological invariant, which is determined by bulk electronic structure (the existence of energy gap). In order to change the topological invariant at the surface of 3D topological insulator the energy gap needs to be closed. This results in states inside the energy gap, crossing the Fermi energy and localized on the surface of the material. It can be shown that in Z2 topological insulators with inversion symmetry the nontrivial topological invariant (ν = 1) arises because of the band inversion at an odd number of high symmetry points in the Brillouin zone. In the Weyl semimetal there is an intrinic twofold degeneracy in the toching point of valence and conduction band (Weyl point). In order to make the bands in Weyl semimetal single-degenerate, the time reversal or inversion symmetry needs to be broken. Weyl points always come in pairs of opposite chirality and can be destroyed only by the annihilation process, which makes them topologically protected. Direct consequences of chirality of Weyl points are surface states in the form of Fermi arcs (can be measured by ARPES method) and chiral anomaly which results in negative magnetoresistance for E parallel to B. In Dirac semimetal inversion and time reversal symmetry are preserved and Dirac points, which can be seen as two Weyl points at the same momenta, are fourfold degenerate. Dirac semimetal can occur at topological phase transition between normal and topological insulator or as a stable intrinsic phase protected by crystalline symmetry (rotational symmetry that prevents hybridization of degenerate states). In this case, Dirac points always come in pairs. If magnetic field is applied in the direction of the axis of the rotational symmetry, Dirac point splits into two Weyl points. Nodal line semimetals are systems in which the conduction and valence bands cross each other along line inside the Brillouin zone and are linearly dispersed perpendicular to the crossing line. Touching lines are again protected by crystalline symmetries (mirror and glide planes). Measuring quantum oscillations is one of the main experimental method for characterization and investigation in topological insulators and topological semimetals. In this section the theory of quantum oscillations is covered in detail. In the strong magnetic field the electron motion is quantized and electron states are located in Landau levels. The number of Landau levels decreases with an increasing magnetic field (levels become more degenerate). By increasing the field Landau levels will cross the Fermi energy. It can be shown that this crossing is periodic in the inverse magnetic field 1/B. This periodic change of the density of sates at the Fermi energy leads to oscillations in almost all physical quantities. Only electrons from extremal cross sections of the Fermi surface and the plane normal to the magnetic field contribute to quantum oscillations and the frequency of oscillations is determined by the surface area of this cross section. Therefore, by measuring quantum oscillations one can determine the parameters of different parts of the Fermi surface. Bi1.1Sb0.9Te2S is a 3D topological insulator with highly reduced bulk carriers density. Monocrystalline samples of Bi1.1Sb0.9Te2S were grown by modified melt crystallization method. The transition from semiconducting to metal behaviour in temperature dependence of the sample resistance below ≈ 100 K, which can be explained by the domination of conductive surface states, was observed in Bi1.1Sb0.9Te2S samples. Strong quantum oscillations in the magnetoresistance of thin Bi1.1Sb0.9Te2S samples, which disappear if the magnetic field lies in the plane of the sample, were observed. This can be an indication that the oscillatory contribution is from conductive surface states only. It was found that the frequency of quantum oscillations in magnetoresistance strongly decreases with increasing temperature (drop of ≈ 50% from 2 - 45 K). It is expected that in Pb1-xSnxSe for x ≈ 0.17, the topological phase transition from normal insulator to topological crystalline insulator takes place. This corresponds to the case of the Dirac semimetal. Magneto-transport and magnetization measurements, in fields up to 16 T and 5 T, were carried out in the synthesized Pb0.83Sn0.17Se samples with reduced carrier density. By quantum oscillation analysis the charge carriers parameters in Pb0.83Sn0.17Seare determined. By determining the phase of quantum oscillations, which is directly related to the electron Berry phase, the existence of Dirac fermions in Pb0.83Sn0.17Se is confirmed. A strong linear magnetoresistance, whose temperature dependence can be scaled by the temperature change in the mobility of carriers, was observed in Pb0.83Sn0.17Se. Cd3As2 is an intrinsic Dirac semimetal with a pair of symmetry protected Dirac points located at rotational symmetry axis. Monocrystalline samples of Cd3As2 were obtained by specially designed chemical vapour deposition technique and characterized by magnetization and magneto-transport measurements. By controlling the synthesis parameters, Cd3As2 samples of significantly different quantum oscillation frequency (Fermi energy), from 55 - 15 T, were synthesized. In the sample of Cd3As2 with the lowest Fermi energy, a magnetic torque was measured in the fields up to 35 T. In the field around the quantum limit (15 T), the angle dependent anomaly was observed in the magnetic torque. This anomaly can be associated to the transition from Dirac to Weyl semimetal, with the magnetic field applied in the direction of rotational symmetry axis in Cd3As2. As Pb0.83Sn0.17Se, Cd3As2 samples also show a strong linear magnetoresistance, whose physical origin is still unknown. A negative magnetoresistance, for B parallel to E, is found in Cd3As2. This can possibly be linked to the effect of chiral anomaly. ZrSiS and HfSiS are nodal line semimetals of the same quasi two-dimensional crystal structure and very similar Fermi surface, which consists of multiple electron and hole Fermi pockets. Monocrystalline samples of ZrSiS and HfSiS were grown by standard chemical vapour deposition technique and characterized by magnetization measurements. A strong quantum oscillations were measured in the magnetic torque of ZrSiS and HfSiS, in a magnetic field up to 35 T along the direction of c crystal axis. It is found that the oscillatory part of magnetic torque consists of low frequency contribution (frequencies up to 1000 T) and high frequency contributions (frequencies from 7 - 20 kT in ZrSiS and from 7 - 10 kT in HfSiS). Some of the low frequencies can be associated to different Fermi pockets from kz = π plane in Brillouin zone. The appearance of high frequency quantum oscillations is explained by the effect of magnetic breakdown, in which electrons, in high magnetic field, gain enough energy to tunnel between separated Fermi pockets. The separation of Fermi pockets depends on spin-orbit coupling which is stronger in HfSiS. This agrees with fact that high frequency oscillations in HfSiS start to appear at nearly two times greater magnetic field than in ZrSiS(25 T in HfSiS, and 13 T in ZrSiS).