Ramen is a noodle dish that was originally imported to Japan from China. Over the last few decades, however, ramen has become a typical Japanese dish and gained great popularity inside and outside of Japan. Ramen noodles are about as thin as spaghetti and are served in a soup with various toppings. There are many different ramen dishes differing in the soup base and the toppings:

Some of the most popular soups:

Shoyu Ramen: Brown, transparent, soya sauce based.
Miso Ramen: Brown, non-transparent, miso based.
Shio Ramen: Transparent, salt based.
Tonkotsu Ramen: White, milky, pork based.
Some ramen dishes with a feature topping:

Chashumen: BBQ pork ramen.
Tanmen: Vegetable ramen.
Wonton ramen: Ramen with small Chinese dumplings.
and many more
My Ramen Picks
Taishoken - basic shoyu ramen - very tasty and served in large bowl (with fingers in the soup). The noodles are thick and the soup taste really good, you'll come back for more ! Taishoken has other shops, but they're not as good as the one in Ikebukuro . . . .
Tsuke-men, another favorite - this cold noodle is heavenly !. The shop is only open from 11:30 to 3 PM, so you'll have to line up early coz sometimes they run out of soup, then the shopmaster will close the shop for the day !
Taishoken is 2 mins from Sunshine City Ikebukuro. You'll see a long line everyday rain or shine (like this one).
Jiro is one of my favorites. They have one shop next station to where I live. As you can see, Jiro is also very popular (and so you'll have to line up to get your fill). For some people (especially those who don't like oily stuff in their broth) this type of ramen may be too much. But then again, it's a matter of personal preference (some may love it and some may not). Obviously, Jiro has its own followers, as shown by the long lines at Jiro at any given time.

The thing is, ramen shops all around Japan have gotten so diverse that there's almost a suitable ramen for everyone's cravings. At Jiro, you can tell the chef to minimize the oil (lard) that they sprinkle on the soup and ask for more servings of the veggies instead (or even some some more garlic). The key to enjoying ramen is probably finding the right one that suits your taste.

Jangara - you get all these toppings for under 1,000 yen. The main shop at Akihabara is a bit cheaper but is always crowded. Expect to wait for at least half an hour or more before finally getting inside the small nook, where you will see the fat chef who looks more like a shaolin master in one of those kung-fu movies. There's also a Jangara near Omotesando. Jangara is by far the best among tonkutsu-based ramen !
Jangara
TAISHOKEN
JIRO
KORYU
The one in Shinjuku is full of people most of the time (mostly young couples and salarymen during the early evenings). The place is nice and the basic ramen they serve is good (it's also a bit cheaper)
Ippudo
Ippudo - I love their akamaru ramen (its spicy and very tasty). The soup is basic tonkutsu (pork broth) and has a deep flavor. Unlike other tonkutsu-based ramen, this one doesnt smell "porky" and the shiromaru is even more mild. Enjoy the sauteed moyashi which is laid out on the table as topping for the noodles (its free, so you can eat it as side dish or together with your favorite Ippudo dish). 
Ippudo Shiromaru ramen
more to follow on next update . . .
SANTOKA
Santoka, is the "prince" of Shio-Ramen. I especially recommend the one in Ebisu.  Santoka is one of the best places if you're into Shio Ramen. By the way, they also have Miso and Shoyu served with tasty "chashu". Don't miss the "Toru Niku" ramen (very soft, roasted pork), which they serve with complimentary onigiri on the side during lunch time.
Ramen Freaks
MUSASHI - Shinjuku
MUSASHI - in Shinjuku is Shoyu ramen. The soup is seafood-based (fish stock) and much like the "tsuyu" that is used with soba (buckwheat noodles). The taste is surprisingly nice and delicious inspite of its simplicity. Expect long lines along the sidewalks (sometimes extending to the next street block!). The ramen they serve is reasonably priced and you'll probably come back for more (if you can stand the long waits)
Check it out !
Shin-Yokohama Ramen Museum (near Shin-Yohama Stadium) is a magnificent replica of old Japan depicting the aura and splendor of the "old days". This is the disneyland of Ramen. Choose from a varitey of ramen in many shapes, sizes, flavours ! Here you'll find some very good ramen shops (they select some of the best ramen shops and put up a small stall here). This one's just a few minutes from Shin-Yokohama station and very near the Arena (stadium) so you won't miss it.
Shin-Yokohama Ramen Museum
Mutekiya - Ikebukuro
Nantsuttei
Saijyo
Kujiraken
New Ramen attraction in Shinagawa
Of the above Ramen shops, Nantsuttei is my favorite !
If you happen to be in Shinagawa, don't miss the Shinatatsu  Ramen Shops near the Shinagawa station (http://www.shinatatsu.com). Here, you will see a long line leading to "Nantsuttei" (see pic above). Nantsuttei is one place I suggest if you have time to spare (the waiting time is 30 minutes, at least). They open at 11:30 and close shop at 22:00.  Its a 2 minute walk from the Takanawa Exit (Takanawa Prince Hotel exit). As soon as you get off the station, you'll see the Prince Hotel in front of you, so just turn left and walk a few meters. You won't miss it (there's lots of people lining up the store front).

Nantsuttei already has a loyal following among the Ramen fanatics (including me). The soup is very tasty and deeply-flavored (slightly brownish) , topped with soft roasted pork and other condiments of your choice. There's fresh garlic for you to crush and also bite-size gyouza (sold separately on the vending machine) that goes well with the broth.  ENJOY !!!
Mensouoka
For those who prefer "Tsuke-men", here's the perfect place to satisfy your cravings. If you're coming from Yokohama, then its just a five-minute ride on the Keikyu line. Get off at Kanagawa Shin-machi station and turn left into the main road. From there it's just a 3 minute walk (there's a Convenience Store beside it, so its easy to spot). The place is dark-lit and there are only counter seats for about 10 people or so at a time. This place is also crowded and people line up to the street corner.

Although they have hot ramen, I would recommend their specialty "Tsuke-men". You pay via the vending machine on the entrance (900 yen for the spicy  tsuke-men). It may be a bit pricey but the taste will be worth the trip and hassle. You'll get a choice of spicy or regular tsuke-men. The thing I like about their tsuke-men is the "chashu" or roasted pork topping. They roast the pork first on hot charcoal (sumi-bi) and you'll get large (thick) servings of it. So far, this is the best Tsuke-men I have ever tried  !..

You'll have to try it to beleive it !
"Misoya" in Jimbocho is my latest find. It's basically Miso Ramen but comes higly recommended by my friend who hails from Hokkaido, who says it taste like the Sapporo Ramen back in his hometown. I've tried other Miso Ramen shops around Tokyo but so far this one's the best I've had. I ordered the regular Miso Ramen with lots of chashu or roasted pork topping (at extra charge, of course). It's very good and for 700 yen (for the regular bowl), it's quite reasonable. If you're around Jimbocho on a book hunt, check this place out for lunch (or dinner). If you're taking Hanzomon Line, take the A5 Exit and turn left till you see an AMPM Convenience Store. It's the small shop (counter seats mostly) on the block. Tthere'll be lots of people in line, so you won't miss it. Enjoy !
Misoya Jimbocho
Sumire is another shop (apparently based in Hokkaido) that is fast gaining some loyal following among Miso Ramen fans. I went to their new shop in Lazona Shopping complex right next to Kawasaki station. It's located on the Food Court, along with other ramen shops. Try the Chashu-Mizo ramen. The soup is piping hot and with deep flavour that I like. Aside from the generous chunk of pork (for Chasu ramen), it also has some tiny  bits of pork with the broth (similar to the pic above).

Lazona Food Center is crowded on the weekends so it's best to go on the weekdays, if possible.
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