The basic building blocks of all biomembranes are lipid molecules, which self-assemble to form a very thin and stable barrier, where a variety of proteins can be incorporated into its structure. These two-dimensional systems exhibit a plethora of physical phenomena, which are an abundant source of inspiration for a physicist. The physical aspects of biomembranes are described within a phenomenological model, the so-called Canhan{Helfrich theory, which relies primarily on the geometrical aspects of the membrane surface at large scales. Using this theory, we study the response of a membrane to the inclusion of a transmembrane protein or a protein coat by coupling the composition to the mean curvature. A transition is found from an overdamped to an underdamped regime for the membrane shape and its compositional variation. This leads to large membrane undulations near the inclusion, resulting in the activity suppression of mechanosensitive channels and a preference for the formation of protein coats. We also re-examine the methodology for inferring the bending modulus of membranes from their observed thermal fluctuations. Particularly, we analyse the effect due to the optical projection of such shape undulations across the focal depth of the microscope. A comparison of this with the literature approaches reveals a systematic decrease in the value of the bending modulus, resolving a previously recognised discrepancy between shape measurements and other known techniques. Lastly, we investigate an non-equilibrium model for the formation of membrane domains that also involves membrane recycling. The dynamics and the steady-state features of the domain size distribution are analytically revealed and the implication to the heterogeneity observed in biomembranes is discussed.