I love Robert Crais Elvis Cole books. I’ve read all of his books and like those the best. Not a Patterson, Lippman, Grafton fan. My absolute favorite detective/mystery series is the Louise Penny Inspector Gamache series. New book in August! She gets better and better and better.
Of those mentioned so far, I second the recommendation of Michael Connelly, John Sanford, and Dennis Lehane. Crais is pretty good. I also love all of Thomas Perry, but particularly Jane Whitefield. Tony Hillerman’s books are first rate, and you can learn a lot about Navaho and Hopi culture.
If you do decide to read Louise Penny, and a lot of people are devoted to her, start at the beginning. I made the mistake of picking up one in the middle of the series, and it was as irritating, as someone remarked, as being at a cocktail party where everyone else knows each other and the conversations all have to do with stuff you don’t know about. 
I’m going to try again one day, with the first book.
A couple of Scottish writers whom I particularly enjoy are Denise Mina and Val Mcdermid. Mina is more gritty, moody, and novelistic. McDermid is more of a page turner, but also very interesting. This is a good overview of Mina’s books, which are all good:
http://www.denisemina.com/books/
Val Mcdermid wrote an early series about a PI named Kate Brannigan that I would skip. Her Tony Hill books are great:
I could go on listing mystery novelists forever, but my personal favorite is Michael Gruber. For a long time, he wrote the Robert K. Tannenbaum series. (Tannenbaum is his cousin.) Eventually, he decided to write his own books, and they got a new ghost writer for Tannenbaum, at which point that series went over a cliff, IMNSHO. I would strongly recommend his first publication under his own name, Tropic of Night. All I could say when I finished it was WOW! It is the first of his Jimmy Paz books, although Paz is not the main character. Gruber is a brilliant guy, and his writing is always on a level above.
For pure page turning enjoyment that at the same time doesn’t insult your intelligence, you can’t beat Lee Child’s Jack Reacher books. I’d try to start at the beginning, with Killing Floor. Ignore the horrendous Tom Cruise in the movie, which nearly caused mass apoplexy in Reacher fans worldwide. One of the important things about Reacher is that he is a huge guy, about 6’5", and, incidentally, blond. The way he uses his size is a significant theme in the books.
Some of my favorites haven’t been mentioned yet: Ann Cleves (if you like Vera or Shetland on PBS, these are the books they are based upon); Peter May, Camilla Lackberg, Maurizio de Giovanni.
Michael Gruber is amazing! I love those Jimmy Paz books - mysteries with an eerie twist. I haven’t read any of his other books but I’m tempted.
I started with book 7 of Louise Penny, and actually thought it worked better for me. I think it gets stronger as it goes, and then was eager to go back & get the backstory.
Venetian atmosphere - Donna Leon
Sicily - Andrea Camilleri
Southwest France - Martin Walker
Of course Louise Penny, Denise Mina (wonderful), Deborah Crombie, Laura Lippman, Tana French, and Laurie King.
You can also go to current and prior years’ nominees and winners of the Edgar awards My local Seattle Mystery Bookshop has a newsletter available on their website which gives descriptions and write ups of new books. Prior issues are archived and are a nice resource.
Can’t believe I forgot Peter May. Loved The Lewis Trilogy. I wish there were more in that series. Gruber is on my list. I also enjoy the Maisie Dobbs and the Flavia de Luce books but I’m not ravenous for them.
+1 on @zeebamom’s recommendations of the Donna Leon and Andrea Camilleri series. It’s fun learning about Italian culture while reading them.
Haven’t had time to look at all the the recommendations, but has anyone mention Elizabeth George? I just read one of her latest - it was good. I read the latest installment of the Girl with the Dragon Tattoo series (a different author) and I liked it too. I read a book called “The Long and Far Away Gone” by a writer from my homestate of OK featuring a fictional depiction of some famous murders - one from OK, then a video store murder in ALBQ (I lived in both places when they happened) - I think I liked the book alot just because of the familiarity of the setting in OKC though. I loved Unbroken. I really like the Flavia de Luce books a couple of years ago. The main character is an 11 year old girl who loves chemistry and solves mysteries in the English? countryside. “Luce lives on the once glorious, now crumbling estate of Buckshaw, in the bucolic English hamlet of Bishop’s Lacey, where murder happens more than it should and the brilliant amateur detective (and dedicated poison enthusiast) spends equal time dodging her older sisters and solving the most ghastly of crimes.” I like the April Woo books by Leslie Glass and I liked the fact that Ms. Glass sent me autographed copies when I ordered a few from her. I liked The Informationist: A Vanessa Michael Munroe Novel. I wish someone would compile a list of all the recommended books so that you could just cut and paste and add your recommendation at the bottom. Three hundred posts is a lot to scroll through. BTW agree with Denise Mina and Louise Penney.
Oh, and I love reading the “Traveller’s Tales” books. Mostly compilations of travel articles from magazines, but really good reads if you like travel.
Margaret Maron’s Deborah Knott series set in NC. It’s always fun to read a book set in a familiar place.
Greg Iles Penn Cage series. I’ve only read two of his books but boy were they good!
My favorite ‘literary mystery’ writers:
Tana French (Dublin Murder Squad)
Kate Atkinson (Jackson Brodie)
Louise Penny (Inspector Gamache)
Dennis Lehane (Patrick/Angie, Shutter Island)
If you don’t mind creepy/disturbing:
Mo Hayder (Jack Caffery)
John Connolly (Charlie Parker)
Denise Mina
I think the super-prolific writers (Sanford, Davenport et al) may have started out well but became terrible.
Rachel Howell Hall: (Detective Elouise Norton)
and
Becky Masterman: (Brigid Quinn)
@LeastComplicated mentioned The Informationist. I liked it also - just … I never see it mentioned.
I also like Alan Bradley’s Flavia de Luce series and Lee Child’s Jack Reacher series (both mentioned in above posts).
I’ve recently fallen in love with Liane Moriarty. Fast-paced, humorous, but psychologically astute. Loved Big Little Lies, and am currently reading the Hypnotist’s Love Story.
Love the Tana French books. Just finished The Dry, which someone on this thread recommended. It was a good one too.
I read the first Greg Iles book The Quiet Game and wasn’t thrilled. Do they get better?
@cartera45, I haven’t read The Quiet Game, but I really enjoyed Iles’ Natchez trilogy.
I read The Quiet Game and then read the second in the Penn Cage series, Turning Angel. On a scale of 1-5 I rate Turning Angel a 1 - omg, sexist and then some. FWIW, the Natchez Trilogy technically comprises books 4-6 of the Penn Cage series. I’ve heard the Natchez books are good but Turning Angel was so not my cuppa that I haven’t continued with Iles.
Just read Eleanor Oliphant is completely fine - devoured it in one day - really enjoyed it
Mo Hayder mentioned above. WOW awesome and super creepy…especially to me her first on Japan.