The London penetration depth has been measured in various doping levels of single crystals of Ba(Fe1-xTx)2As2 (T=Co,Ni,Ru,Rh,Pd,Pt,Co+Cu) superconductors by utilizing a tunnel diode resonator (TDR) apparatus. All in-plane penetration depth measurements exhibit a power law temperature dependence, indicating the existence of low-temperature, normal state quasiparticles all the way down to the lowest measured temperature, which was typically 500 mK. Several different doping concentrations from the Ba(Fe1-xTx)2As2 (T=Co,Ni) systems have been measured and the doping dependence of the power law exponent, n, is compared to results from measurements of thermal conductivity and specific heat. In addition, a novel method has been developed to allow for the measurement of the zero temperature value of the in-plane penetration depth by using TDR frequency shifts. By using this technique, the doping dependence of the zero temperature value of the penetration depth has been measured in the Ba(Fe1-xCox)2As2 series, which has allowed also for the construction of the doping-dependent superfluid phase stiffness. By studying the effects of disorder on these superconductors using heavy ion irradiation, it has been determined that the observed power law temperature dependence likely arises from pair-breaking impurity scattering contributions, which is consistent with the proposed s+- symmetry of the superconducting gap in the dirty scattering limit. This hypothesis is supported by the measurement of an exponential temperature dependence of the penetration depth in the intrinsically clean LiFeAs, indicative of a nodeless superconducting gap.