We obtained spectra, in the wavelength range lambda = 995 - 1769 nm, of all four known planets orbiting the star HR 8799. Using the suite of instrumentation known as Project 1640 on the Palomar 5-m Hale Telescope, we acquired data at two epochs. This allowed for multiple imaging detections of the companions and multiple extractions of low-resolution (R ~ 35) spectra. Data reduction employed two different methods of speckle suppression and spectrum extraction, both yielding results that agree. The spectra do not directly correspond to those of any known objects, although similarities with L and T-dwarfs are present, as well as some characteristics similar to planets such as Saturn. We tentatively identify the presence of CH_4 along with NH_3 and/or C_2H_2, and possibly CO_2 or HCN in varying amounts in each component of the system. Other studies suggested red colors for these faint companions, and our data confirm those observations. Cloudy models, based on previous photometric observations, may provide the best explanation for the new data presented here. Notable in our data is that these presumably co-eval objects of similar luminosity have significantly different spectra; the diversity of planets may be greater than previously thought. The techniques and methods employed in this paper represent a new capability to observe and rapidly characterize exoplanetary systems in a routine manner over a broad range of planet masses and separations. These are the first simultaneous spectroscopic observations of multiple planets in a planetary system other than our own.