They aren’t quite cozy, but they aren’t quite police procedural novels. The detectives involved may be amateur or professional, and they span a couple of centuries. Here are a few to get you started.

The Father Brown Mysteries by G.K. Chesterton are compilations of short stories. The first volume, compiled in 1911, is called The Innocence of Father Brown. There have been further stories and television series based on the Catholic priest, many written by others after Chesterton’s death.
The Lord Peter Wimsey Mysteries by Dorothy L. Sayers, featuring the suave Lord Peter and his love interest Harriet Vane, are a treasure. My friend Elaine first introduced me to them. At that time, probably about fifteen years ago, the only copies I could find were old, worn editions on the shelves of my public library – my favorite kind of book. The first book in this series, which takes place in England during the time period between the two World Wars, is Whose Body. There are eleven books in the original series, which was picked up and continued more recently by Jill Paton Walsh.
Sir Arthur Conan Doyle’s Sherlock Holmes is iconic in classic crime fiction/mysteries. Believe it or not there are only four Holmes novels (the first is A Study in Scarlet), but there are many more short stories about the man. Although they were written in the 1880s, Holmes and his sidekick (and frequent narrator) Dr. Watson seem to have taken on a life of their own, with multiple films and television shows based on the characters. We all recognize the hat, right?
The Adam Dalgliesh books by P.D. James are written in (relatively) present day. Dalgliesh, of the New Scotland Yard, solves murders but is also a poet – an introspective man – perhaps a melancholy version of Armand Gamache. James writes richly detailed, intelligent crime fiction, and each of these novels is engrossing enough that you will feel a bit wistful as you finish the last in the series, which is complete now as P. D. James is deceased. The first book in the series is Cover Her Face.
The Moonstone, by Wilkie Collins, is arguably the first detective fiction written. The 1868 novel written about a stolen diamond takes place in Britan, but it has a distinct Indian influence as the diamond, inherited by one of the main characters, was mined in India and is considered sacred by the Hindu priests who would like to recover it. A sense of magic pervades the tale, which is worth a read.