## 1 Introduction  
The research field on materials for gas separation membranes is constantly expanding due to the pressing industrial request for more performing materials to employ in gas treatment, such as hydrogen recovery (H2/CO2) [1], oxygen-enriched air production (O2/N2) [2, 3], biogas up-grading [4, 5] and natural gas treatment (CO\({}_{2}\)/CH\({}_{4}\)) [6], and post-combustion carbon capture from flue gas (CO\({}_{2}\)/N\({}_{2}\)) [7]. This increasing interest is dictated by the advantages of membrane technology compared to traditional gas separation techniques. Gas separation by means of membrane technology is an economical process; it is easily scalable and it can be used in non-drastic temperature and pressure conditions, which are more environmentally friendly. Despite numerous efforts to develop new materials for gas separation, the membrane market still needs to overcome some challenges. In fact, the highly permeable rubbery polymers present low selectivity, and the highly-selective glassy polymers are less permeable. To overcome this trade-off between permeability and selectivity, extensively reported by [PERSON] in 1991 and in 2008 [8; 9], researchers are focusing on the design and development of hybrid membranes based on the combination of two different materials in order to get the advantages of both [10]. According to this concept, a valid strategy is the embedding of porous metal-organic frameworks (MOFs) in the polymer matrix in order to obtain mixed matrix membranes (MMMs) with enhanced gas transport properties [11; 12; 13; 14; 15; 16]. MOFs are an attractive new class of microporous materials built by the combination of metal atoms/clusters with a wide variety of organic ligands, which can be specifically designed for improving the compatibility with the organic polymer phase and with high specificity for gases [17; 18]. The gas transport in polymer membranes occurs following the solution-diffusion mechanism in which permeability and selectivity are determined by kinetic parameters (diffusion) and thermodynamic factors (solubility), via \(P=D*S\); \(\alpha_{\frac{k}{y}}=\frac{D_{x}}{Dy}*\frac{S_{x}}{Sy}\). The addition of MOFs can improve the gas transport properties of the neat polymer by influencing these two parameters [10]. The diffusion, being a kinetic phenomenon, is strongly linked to the free volume of the polymer materials and to the molecular size of the penetrating species. In the MOFs, the link between metals and organic units forms a three-dimensional structure with cages having well-defined shape and size, which can improve free volume elements of a matrix, increasing gas transport in terms of permeability. At the same time, the cages of MOF forming a preferential pathway for a specific gas can act as a molecular sieve, increasing membrane selectivity. On the contrary, the main factor that determines the solubility in a polymer matrix is the ability of the penetrant gases to condense, which in turn is correlated with the interactions that occur between the gas and the matrix of membrane. MOFs chemical structure can be easily designed or functionalized synthetically in order to improve its affinity for specific gases [19]. In this case, the membrane permeability is expected to improve as a consequence of an increased contribution in solubility due to the gas condensability in the MOF. Furthermore, the introduction of chemical groups with a specific affinity to one gas in a mixture will also increase its selectivity. The choice of the materials combination for MMMs preparation must be made on the basis of specific physical and chemical properties, in order to tailor in advance the membrane for a desired gas separation. For this reason, it is necessary to understand how the addition of MOF can influence the gas transport properties of different polymers, rubbery or glassy.