In the first part of my thesis, I report the discovery that the refractive index sensitivities of subwavelength plasmonic sensors obey a universal linear scaling property, in which the sensitivities versus resonant wavelength exhibit a slope equal to 1/RIU (RIU = refractive index unit) instead of 2/RIU predicted by earlier theory.Similar scaling relation also applies to plasmonic molecules of diverse geometrical structures, coupling strengths, distributions and intensities of electromagnetic hot spots. The universal scaling relation reveals the fundamental standing wave resonances for all plasmonic atoms and the predominant near-field electric couplings for most plasmonic molecules investigated so far. We will also propose a method of detecting surface tension at nanoscale using complementary plasmonic structures. In the second part, I further employ electromagnetic simulation to study the fundamental electromagnetic duality of various plasmonic atoms by testing their sensitivities to the environmental permeability changes. Interestingly, we find the electromagnetic duality is broken in various plasmonic atoms with respect to the environmental permittivity and permeability sensors. In contrast to the universal behavior of permittivity sensitivities, the permeability sensitivities deviate from the straight line with decreasing resonant wavelengths and are strongly geometrically dependent. Moreover, for a given plasmonic atom, we can always find a particular resonant wavelength below which the permeability sensitivities drop to zero. We notice that the electromagnetic duality is strongly correlated with the effective plasma frequency of the plasmonic atoms and thus they can be further manipulated by introducing the concept of spoof plasmons. Unlike traditional electromagnetic systems such as microwave waveguides or antennas, our results provide a rare example that the fundamental electromagnetic duality can be broken in plasmonic systems. Finally, the property of multiply-connected, topological metamaterial and several designed plasmonic dimer will be discussed in remaining parts of the thesis including methods for electromagnetic parameters retrieval, investigations on breakdown of standing wave model, and coupling effects of plasmonic resonance that deviate from the traditional understandings.