We report the discovery of four transiting extrasolar planets (HAT-P-34b-HAT-P-37b) with masses ranging from 1.05 to 3.33M_J_ and periods from 1.33 to 5.45days. These planets orbit relatively bright F and G dwarf stars (from V=10.16 to V=13.2). Of particular interest is HAT-P-34b which is moderately massive (3.33M_J_), has a high eccentricity of e=0.441+/-0.032 at a period of P=5.452654+/-0.000016days, and shows hints of an outer component. The other three planets have properties that are typical of hot Jupiters.