This thesis work contains electrical transport and low frequency (1/f) noise measurements in ultrathin graphene and Molybdenum disulphide (MoS2) field effect transistors (FET). From the measurements, We mainly focus on the origin of disorder in both the materials. To address the orgin of disorder in graphene, we study single and bilayer graphene-FET devices on SiO2 substrate. We observe that both conductivity and mobility are mainly determined by substrate induced long range, short range, and polar phonon scattering. For further confirmation, we fabricate suspended graphene devices which show extremely high mobility. We find that, in contrast to substrate-supported graphene, conductivity and mobility in suspended graphene are governed by the longitudinal acoustic phonon scattering at high temperature and the devices reach a ballistic limit at low temperature. We also conduct low frequency 1/f noise measurements, known to be sensitive to disorder dynamics, to extract more information on the nature of disorder. The measurements are carried out both in substrate-supported and suspended graphene devices. We find that 1/f noise in substarted graphene is mainly determined by the trap charges in the SiO2 substrate. On the other hand, noise behaviour in suspended graphene devices can not be explained with trap charge dominated noise model. More-over, suspended devices exhibit one order of magnitude less noise compared to graphene on SiO2 substrate. We believe noise in suspended graphene devices probably originate from metal-graphene contact regions. In the second part of our work, We present low temperature electrical transport in ultrathin MoS2 fields effect devices, mechanically exfoliated onto Si/SiO2 substrate. Our experiments reveal that the electronic states in MoS2 are localized well up to the room temperature over the experimentally accessible range of gate voltage. This manifests in two dimensional (2D) variable range hopping (VRH) at high temperatures, while below ~ 30 K the conductivity displays oscillatory structures in gate voltage arising from resonant tunneling at the localized sites. From the correlation energy (T0) of VRH and gate voltage dependence of conductivity, we suggest that the charged impurities are the dominant source of disorder in MoS2. To explore the origin of the disorder, we perform temperature dependent I - V measurements at high source-drain bias. These measurements indicate presence of an exponentially distributed trap states in MoS2 which originate from the structural inhomogeneity. For more detailed investigation, we employ 1/f noise which further confirms possible presence of structural disorder in the system. The origin of the localized states is also investigated by spectroscopic studies, which indicate a possible presence of metallic 1T-patches inside semiconducting 2H phase. From all these evidences, we suggest that the disorder is internal, and achieving high mobility in MoS2 FET requires a greater level of crystalline homogeneity.