I vividly remember this show as a kid. It had great music, as noted by others, and had a certain grittiness about it that was compelling. Don Collier had the perfect face for a US Marshal -- tough, weary, cynical, and (occasionally) smiling. That he was not handsome in a typical Hollywood way helped the series. He looked like he'd really just spent the last few days in the saddle on the trail of outlaws. Collier played the role of lawman in a serious and responsible way like all westerns back then, but somehow more realistically. I also remember the show for great shoot-out scenes, usually at the end of the show. TV westerns in the early 60s were going through a brief period of especially graphic shoot-outs, and Outlaws benefited from this. I don't mean blood or screams of pain, of course, but a lot of grazing bullets and fancy gunplay. The bad guys when shot would crash very forcefully through windows behind them, or would flip over the hitching post rail, etc. Great time for stuntmen. Great time to be a kid watching this show. I'd love to see it again.
