Using a high-resolution differential technique we have determined the electronic specific heat coefficient γ (T ) = C el /T of Ba 1−x K x Fe 2 As 2 with x = 0 to 1.0, at temperatures (T ) from 2 K to 380 K and in magnetic fields H = 0 to 13 T. In the normal state γ n (x,T ) increases strongly with x at low temperature, compatible with a mass renormalization ∼12 at x = 1, and decreases weakly with x at high temperature. A superconducting transition is seen in all samples from x = 0.2 to 1, with transition temperatures and condensation energies peaking sharply at x = 0.4. Superconducting coherence lengths ξ ab ∼ 20Å and ξ c ∼ 3Å are estimated from an analysis of Gaussian fluctuations. For many dopings we see features in the H and T dependencies of γ s (T ,H ) in the superconducting state that suggest superconducting gaps in three distinct bands. A broad "knee" and a sharp mean-field-like peak are typical of two coupled gaps. However, several samples show a shoulder above the sharp peak with an abrupt onset at T c,s and a T dependence γ s (T ) ∝ 1 − T /T c,s . We provide strong evidence that the shoulder is not due to doping inhomogeneity and suggest it is a distinct gap with an unconventional T dependenceWe estimate band fractions and T = 0 gaps from three-band α-model fits to our data and compare the x dependencies of the band fractions with spectroscopic studies of the Fermi surface.The electronic specific heat γ measured to room temperature contains a wealth of quantitative information about the electronic spectrum of metallic systems over an energy region ±100 meV about the Fermi level, crucial for understanding high-temperature superconductivity. Measurements of the electronic specific heat have played an important role in revealing key properties of the copper-oxide-based "cuprate" high-temperature superconductors (HTSCs). Some examples include the normal-state "pseudogap," 1-4 the bulk sample inhomogeneity length scale, 5 and more recently evidence that the superconducting transition temperature is suppressed due to superconducting fluctuations. 6 In this work we extend such measurements to the iron-arsenide-based "pnictide" HTSCs. Here we present results obtained for polycrystalline samples of Ba 1−x K x Fe 2 As 2 (x = 0 to 1.0) using a high-resolution differential technique. 7 With this technique we directly measure the difference in the specific-heat capacities of a doped sample and an undoped reference sample (BaFe 2 As 2 ). This eliminates most of the large phonon term from the raw data and yields a curve dominated by the difference in electronic terms. Features of the electronic specific heat that would otherwise be masked by the large phonon background are then clearly visible in the raw data over the entire temperature range. Central to the success of this technique are measurements on a series of samples at closely spaced doping intervals, so that systematic trends in the the relatively small difference in phonon terms between sample and reference can be identified and appropriate corrections made. ...