Two-dimensional layered materials, such as graphene and MoS2, are emerging as an exciting material system for a new generation of atomically thin electronic devices. With their ultrahigh surface to volume ratio and excellent electrical properties, 2D-layered materials hold the promise for the construction of a generation of chemical and biological sensors with unprecedented sensitivity. In my PhD thesis, I mainly focus on graphene based electronic biosensors and chemical sensors. In the first part of my thesis, I demonstrated the fabrication of graphene nanomesh (GNM), which is a graphene thin film with a periodic array of holes punctuated in it. The periodic holes introduce long periphery active edges that provide a high density of functional groups (e.g. carboxylic groups) to allow for covalent grafting of specific receptor molecules for chemical and biosensor applications. After covalently functionalizing the GNM with glucose oxidase, I managed to make a novel electronic sensor which can detect glucose as well as pH change. In the following part of my thesis I demonstrate the fabrication of graphene-hemin conjugate for nitric oxide detection. The non-covalent functionalization through π-π stacking interaction allows reliable immobilization of hemin molecules on graphene without damaging the graphene lattice to ensure the highly sensitive and specific detection of nitric oxide. The graphene-hemin nitric oxide sensor is capable of real-time monitoring of nitric oxide concentrations, which is of central importance for probing the diverse roles of nitric oxide in neurotransmission, cardiovascular systems, and immune responses. Our studies demonstrate that the graphene-hemin sensors can respond rapidly to nitric oxide in physiological environments with sub-nanomolar sensitivity. Furthermore, in vitro studies show that the graphene-hemin sensors can be used for the detection of nitric oxide released from macrophage cells and endothelial cells, demonstrating their practical functionality in complex biological systems. In the last part of my thesis, I demonstrate the construction of few-layer molybdenum disulfide (MoS2) based field-effect transistor (FET) device for highly sensitive detection of Hg2+ ion in aquatic solutions. The detection of mercury in aquatic environment is of great importance because mercury is an environment pollutant with severe toxicity. High binding affinity between mercury and sulfur makes MoS2 a promising candidate for mercury sensing. Our studies demonstrate that MoS2 sensors can selectively respond to Hg2+ ion with a detection limit of 30 pM. This MoS2 FET based mercury sensor promises great potential for highly sensitive, label-free, low-cost, fast and non-aggressive detection of mercury in aquatic environment.