It is well known from Mermin-Wagner theorem that a two dimensional(2D) system with continuous symmetry can have no long-range order at finite temperature. However such systems can undergo a transition from a low temperature phase with quasi-long range order to a disordered phase at high temperatures. This is known as Berezinskii Kosterlitz Thouless (BKT) transition. The BKT transition is characterized by the presence of bound vortex pairs at low temperature which dissociate into free vortices above the critical temperature and has been observed in thin superconducting films, 2D superfluids, 2D liquid crystals etc. In this thesis work, we have used resistance/current fluctuations (low frequency/shotnoise) as a probe to investigate the BKT transition in different 2D systems. This work can be divided into three parts: In the first part, we probe the ground state of interacting electrons in 2D in the presence of disorder. We show that at low enough temperatures (~ 270mK),the conductivity tends to zero at a nonzero carrier density with a BKT-like transition. Our experiments with many two dimensional electron systems in GaAs/AlGaAs heterostructures suggest that the charge transport at low carrier densities is due to the melting of an underlying ordered ground state through proliferation of topological defects. Independent measurement of low-frequency conductivity noise supports this scenario. In the second part, we probe the presence of long-range correlations in phase fluctuations by analyzing the higher-order spectrum of resistance fluctuations in ultrathin NbN superconducting films. The non-Gaussian component of resistance fluctuations is found to be sensitive to film thickness close to the transition, which allows us to distinguish between mean field and BKT type superconducting transitions. The extent of non-Gaussianity was found to be bounded by the BKT and mean field transition temperatures and depends strongly on the roughness and structural inhomogeneity of the superconducting films. In the final part of the thesis, we explore the transport mechanism in disordered 2D superconductors using shot noise. The resistivity shows an activated transport in the patterned ultrathin films of NbN at low temperatures signifying the presence of large scale inhomogeneities in the sample. The measurement of current fluctuations yield a giant excess noise at low temperatures which eventually decreases below the measurement background at a temperature corresponding to the normal state of the original sample(before patterning). We attribute the enhancement in the shot noise to a possible occurrence of multiple Andreev reflections occurring in a network of SNS(superconductor-normal-superconductor) junctions formed due to the interplay of disorder and superconducting fluctuations.