The imaging and spectral performance of CdTe double-sided strip detectors (CdTe-DSDs) was evaluated for the ASTRO-H mission. The charcterized CdTe-DSDs have a strip pitch of 0.25 mm, an imaging area of 3.2 cm×3.2 cm and a thickness of 0.75 mm. The detector was successfully operated at a temperature of -20°C and with an applied bias voltage of 250 V. By using two-strip events as well as one-strip events for the event reconstruction, a good energy resolution of 2.0 keV at 59.5 keV and a sub-strip spatial resolution was achieved. The hard X-ray and gamma-ray response of CdTe-DSDs is complex due to the properties of CdTe and the small pixel effect. Therefore, one of the issues to investigate is the response of the CdTe-DSD. In order to investigate the spatial dependence of the detector response, we performed fine beam scan experiments at SPring-8, a synchrotron radiation facility. From these experiments, the depth structure of the electric field was determined as well as properties of carriers in the detector and successfully reproduced the experimental data with simulated spectra.