Putter heads and putters include one or more of: (a) a main putter body portion including a first arm and a second arm; (b) a first polymeric and/or damping element engaged with the first arm; and (c) a second polymeric and/or damping element engaged with the second arm. The putter heads further may include polymeric material within the putter head structure and exposed at a central ball striking portion of the ball striking face, e.g., in elongated openings provided in the ball striking face. Grooves may be provided in the material of the ball striking face and/or in the exposed polymeric material to affect the ball launch. The putter heads may have various additional features or structures, e.g., relating to weighting features, alignment aid features, putter head constructions or parts, polymeric material exposure features, groove features, etc.