Abstract: This piece of work is twofold. First, the time evolution of wave-packets in 2D systems is analyzed by the Split-Operator technique in three different scenarios: in multilayer phosphorene, the transient oscillations in the time-dependent average of position and momentum were observed due to the zitterbewegung effect, and the wave packet propagates non-uniformly along the space deforming itself into an elliptical shape. These results were corroborated by the Green’s function formalism except for large values of the wave-vector and long times; in 2D semiconductor quantum wires (QWs) with anisotropic effective masses and different angle orientations with respect to the anisotropic axis. We have shown that the greater this angle, the smaller is the energy levels spacing implying in an increase of the accessible electronic states. Additionally, for non-null magnetic field, the quantum Hall edge states are significantly affected by the edge orientation. In the anisotropic case damped oscillations in the average values of velocity in both x and y directions where obtained. Theses oscillations are originated by the QW geometry but also from subwavepackets with different momentum orientations, whereas for isotropic QWs the wavepacket disperses without splitting; in the third scenario the split-operator technique was used to study the Landau levels, the wave packet trajectories and velocities of electrons in graphene at low-energy regime described by a modified Dirac equation where the momentum-operator is written in a generalized form as result of applying the position-dependent translation operator formalism (PDTO). In the second part of this thesis, the electronic and tunneling properties of α − T3 lattices were studied. Electrons in these lattices behave analogous to integer-spin Dirac Fermions. The presence of a third atomic site in the unit cell leads to a flat band in the energy spectrum, providing unique electronic and tunneling properties. The presence of a super-periodic potential and the inclusion of symmetry-breaking terms results in deviations of the atomic equivalence between the atomic sites affecting the Dirac points and the band-gap. Small deviations in the equivalence between the atomic sites and the number of barriers change the transmission properties in these lattices. Additionally, new tunneling regions are possible by adjusting the symmetry between the atomic sites and affect the omnidirectional total transmission called super-Klein tunneling observed in these lattices. We compare those results to the tunneling probabilities through regions where the energy spectrum changes from linear with a middle flat band to a hyperbolic dispersion.