We present results of simulations of the climate of the newly discovered planet Proxima Centauri B, performed using the Met Office Unified Model (UM). We examine the responses of both an `Earth-like atmosphere and simplified nitrogen and trace carbon dioxide atmosphere to the radiation likely received by Proxima Centauri B. Additionally, we explore the effects of orbital eccentricity on the planetary conditions using a range of eccentricities guided by the observational constraints. Overall, our results are in agreement with previous studies in suggesting Proxima Centauri B may well have surface temperatures conducive to the presence of liquid water. Moreover, we have expanded the parameter regime over which the planet may support liquid water to higher values of eccentricity (>= 0.1) and lower incident fluxes (881.7 Wm-2) than previous work. This increased parameter space arises because of the low sensitivity of the planet to changes in stellar flux, a consequence of the stellar spectrum and orbital configuration. However, we also find interesting differences from previous simulations, such as cooler mean surface temperatures for the tidally-locked case. Finally, we have produced high resolution planetary emission and reflectance spectra, and highlight signatures of gases vital to the evolution of complex life on Earth (oxygen, ozone and carbon dioxide).