We report the results of searches for strong gravitational lens systems in the Dark Energy Survey (DES) Science Verification and Year 1 observations. The Science Verification data span approximately 250 sq. deg. with a median i-band limiting magnitude for extended objects (10σ) of 23.0. The Year 1 data span approximately 2000 sq. deg. and have an i-band limiting magnitude for extended objects (10σ) of 22.9. As these data sets are both wide and deep, they are particularly useful for identifying strong gravitational lens candidates. Potential strong gravitational lens candidate systems were initially identified based on a color and magnitude selection in the DES object catalogs or because the system is at the location of a previously identified galaxy cluster. Cutout images of potential candidates were then visually scanned using an object viewer and numerically ranked according to whether or not we judged them to be likely strong gravitational lens systems. Having scanned nearly 400,000 cutouts, we present 374 candidate strong lens systems, of which 348 are identified for the first time. We provide the R.A. and decl., the magnitudes and photometric properties of the lens and source objects, and the distance (radius) of the source(s) from the lens center for each system.