I don’t know if this counts because it’s not Western literature, but I’ve been reading epic Chinese classics the last couple of years. First, I read Journey to the West. A few weeks ago I finished Romance of the Three Kingdoms. Now, I’ve just started A Dream of Red Mansions. I’ve also got Outlaws of the Marsh already lined up, which will complete my reading of the four classics of Chinese literature.
Hmm, I am curious now. What was the OP’s definition of “classics”? It seems that we have been very free with the term. We have a lot of “should be classics” or “will be classics.” There are “classical classics” and “books they make you read in high school.” We have some “eastern classics.” Does the phrase classics mean Good Books? After giving my two cents I realized that I strayed from the literal meaning of works from texts from ancient Greece and Rome. And it seems most people have answered in kind.
I do have another read I love and completely recommend. If not a western classic, The Tale of Genji by Lady Murasaki should be read by all. It is in the public domain so you can find it online. It is one of the oldest, if not the first, novels in the world. I remember being enraptured with the story when I read it in high school. It is worth a looksie.
Chaucer’s Canterbury Tales. I still have my phonebook sized hardback edition in the original Middle English from when I had a class on it in uni. Makes a good burglar deterrent: “Oy! You! Put that down 'fore I knocks yer brains out with this!” 
Count of Monte Cristo was the only novel I took with me on a monthlong trip to Australia. I read that thing to tatters.
The Odyssey and the Iliad, natch. I read the first one as a small child and was fascinated ever after by Greek mythology. I also read every mythology book I could get my hot little hands on - Greek, Roman, Celtic, Native American (of various stripes), Chinese, Japanese, a little Aus. Aborigine, etc.
The Lord of the Rings. I was about nine when I shocked my uncle-to-be when he saw me reading the Fellowship of the Ring on my own. I still have a couple editions of it around here.
Tolstoy’s The Coming of the King. Read in high school when I was on an Arthurian kick, loved it to bits, especially the way he mixed pre-Christian and Christian cultures.
Poe, Jack London, Saki, O. Henry… Asimov, Clarke… Every author N. Sane mentioned (you, sir/ma’am, have excellent taste).