Batman Films

We have had a 20th century tetralogy:

Batman (1989)
Batman Returns (1992)
Batman Forever (1995)
Batman And Robin (1997)

and a 21st century trilogy:

Batman Begins (2005)
The Dark Knight (2008)
The Dark Knight Rises (2012)

However, I found Batman Returns both unpleasant and incoherent so I would prefer it to be dropped from the canon. That would leave us with two trilogies. The proposed 20th century trilogy is quite coherent. The films respectively introduce Batman, Robin and Batgirl, although that suggests that the second and third titles ought to be reversed and, in fact, Batman Forever would have made a good concluding title.

The opening of Batman And Robin had strong continuity with previous films:

Robin was played by the same actor as in the previous film;
Alfred Pennyworth and Police Commissioner James Gordon were played by the same actors as in the two previous films;
Batman was not but his mask concealed the entire area of his face;
Batman referred to Superman who had had a film series;
the reference to Superman reinforced Robin's reference in the previous film to the city of Metropolis.

(Marvel have gone much further in bringing their characters together in films.)

Batman And Robin contained the human warmth that was lacking from Batman Returns:

as far as I can remember, unless I am confusing films, we saw Alfred comforting the bereaved child Bruce;
we saw real affection between Bruce and Alfred, transcending their traditional, formal master-servant relationship;
further family feeling was invoked by introducing (this version of) Batgirl as Alfred's niece and as a welcome addition to the Batman-Robin team;
Batman learned to trust Robin and to cut out remarks like "This is why Superman works without a partner";
one of the villains was persuaded to do something good for someone else at the end - Batman villains are usually irredeemable, as in Batman Returns.

Thus, what I see here is a good 21st century trilogy and, potentially, a good 20th century trilogy.