Fuels of the furan family, i.e. furan itself, 2-methylfuran (MF), and 2,5-dimethylfuran (DMF) are being proposed as alternatives to hydrocarbon fuels and are potentially accessible from cellulosic biomass. While some experiments and modeling results are becoming available for each of these fuels, a comprehensive experimental and modeling analysis of the three fuels under the same conditions, simulated using the same chemical reaction model, has - to the best of our knowledge - not been attempted before. The present series of three papers, detailing the results obtained in flat flames for each of the three fuels separately, reports experimental data and explores their combustion chemistry using kinetic modeling. The first part of this series focuses on the chemistry of low-pressure furan flames. Two laminar premixed low-pressure (20 and 40 mbar) flat argon-diluted (50%) flames of furan were studied at two equivalence ratios (phi = 1.0 and 1.7) using an analytical combination of high-resolution electron-ionization molecular-beam mass spectrometry (EI-MBMS) in Bielefeld and gas chromatography (GC) in Nancy. The time-of-flight MBMS with its high mass resolution enables the detection of both stable and reactive species, while the gas chromatograph permits the separation of isomers. Mole fractions of reactants, products, and stable and radical intermediates were measured as a function of the distance to the burner. A single kinetic model was used to predict the flame structure of the three fuels: furan (in this paper), 2-methylfuran (in Part II), and 2,5-dimethylfuran (in Part III). A refined sub-mechanism for furan combustion, based on the work of Tian et al. [Combust. Flame 158 (2011)756-773] was developed which was then compared to the present experimental results. Overall, the agreement is encouraging. The main reaction pathways involved in furan combustion were delineated computing the rates of formation and consumption of all species. It is seen that the predominant furan consumption pathway is initiated by H-addition on the carbon atom neighboring the O-atom with acetylene as one of the dominant products.