"The Untouchables" was one of my favorite shows as a kid. My parents frowned on it but I snuck into the basement to watch it on the TV set deemed too old for the living room.
Other reviewers here have done excellent jobs at capturing the main strengths. I'd like to add extra to two items.
The use of music was powerful. Nelson Riddle wrote loose jazz that was appropriate for the series era as well as the broadcast years. After a while I was as aware of the music as the storyline. 50's and 60's dramas had this trait but none better than "The Untouchables." It drew the viewer in with an iron grip.
The on-screen violence paled in comparison to the menace of the characters and their stories. The Santa Claus episode was one of the most harrowing of the series. No glamour in being a junkie. It did a job on a young mind. And it holds up even in these jaded times. Raw stuff.
It built on a rich history of crime drama and, as with "The Godfather," became a reference for what followed.