While he was a flawed man, I really did like Robert Crawley. The Earl of Grantham is the center of the story, as the whole family revolves around his lordship and its history and future legacy. His dynasty is put into doubt when his heir Patrick dies in the Titanic sinking. Patrick was engaged to Lady Mary (his second cousin) as a political marriage of sorts.
Mainly because Robert's wife Cora is American, and a daughter of a family that has a fortune. At the time of Robert's youth, British lords didn't get a lot of things by right alone, so many of them married American women with wealthy fathers. His father tied to the fortune to the estate, making his heir also the heir to Cora's legacy of the marriage. Which would have been good if Robert had a son, but a girl couldn't be the next Earl.
So he hopes that his new heir Matthew would like one of his daughters, and Matthew falls in love with Mary and eventually marries her. I liked how Robert treated Matthew like the son he never had. Though the two quarreled a little, especially when Matthew bails the estate out and has an equal claim over the mansion. Lord Robert realizes he doesn't like someone else trying to run the show.
While he was bitter and angry towards Tom Branson, I loved how he accepted Tom as a son after Sybil's death. While Robert was good to most of the servants, he had a few moments of being a total jerk, but deep down, he had a good heart.
I thought it was interesting that he actually wanted to fight in World War I. He took the government's offer to rejoin the army, which he is disappointed to learn is a honorary post in a effort to boost the morale of the British citizens at home.
I also thought it was sweet how much he loved his grandchildren, but he seemed to favor Sybbie the most, though the other two could barley speak for most of the series. It hurt him deeply when Sybbie moved to America for a little while, before Tom moved back to England.
I honestly thought Robert would die in the series finale, but he managed to survive. I think Dan Stevens leaving the show early probably changed this. I always imagined when I first started the show, the last episode would have Matthew as the Earl, but that role was destined for his son George.