[BWTraveller]

Well for starters there's Assassination Classroom. I was reluctant to watch this one, but when I did I was hooked. My best friend refused for months of pestering before consenting to one episode, and at the end the only thing he said was "MORE". Immediately prior to the beginning of the series, a huge explosion obliterates 70% of the moon, leaving it a permanent crescent. A couple days later a huge, vaguely octopus-like creature shows up with a government entourage at a classroom for the dregs of a certain school, tells them he's the guy responsible for the moon and that he plans to do the same to Earth in March, but until then he's going to be their teacher. The government agent with him tells the kids that they'll be given a huge sum if they can kill him, a task that even the combined efforts of numerous national militaries can't do. The agent becomes their PE teacher and they spend the next year balancing studying for high school entrance exams and studying ways to kill the unkillable.

A little obscure and a real hit-or-miss, but you might enjoy MAR. A weak guy who's always getting bullied and whose father disappeared mysteriously years ago is suddenly transported to another world where the different environment makes him exceptionally strong. He obtains a powerful, sentient and rather rude weapon shaped like a kendama and begins travelling the world, uncovering the mysteries of why he was summoned and gaining allies along the way, including a girl who looks exactly like his childhood friend and whose experiences apparently turn up in said childhood friends dreams every night. Some aspects look weak and childish, especially the character designs. However, no one in this world is actually playing around, as one of the heroes demonstrates when during a tournament set up by the seriously evil villains of the world he unhesitatingly kills his opponent.

As for possibly less-popular titles that I for one loved...
Elemental Gelade: Cou is a young "Sky Pirate" who can never seem to do anything right, until one day he meets Ren. They live in a world with beings called Edel Raids, which look like human women but live for centuries and have the ability to bond with a selected individual, at which point they become a powerful magical weapon. Ren is the last of a line of incredibly powerful Edel Raids and many people want to get their hands on her, and Cou desperately tries to protect her while escorting her to the Edel Garden, where someone is apparently calling her.

Tower of Druaga: Long ago the great hero Gilgamesh had climbed a tower and slew the dark god Druaga to save his beloved Ki, but Druaga placed a curse on him. Now, a new tower has formed filled with monsters, at the top of which stands the resurrected Druaga and the Blue Crystal Rod, an ancient artifact that can grant any one wish. Jil is a climber with lofty dreams of slaying the demon god and becoming a hero, but with his experience it's nothing but a pipe dream and his older brother's party quickly abandons him. Despite this, a young oracle named Kaaya invites him to join her party and help escort her to the top of the tower.

Vision of Escaflowne: Hitomi, a young high school girl with some manner of slight precognitive power is suddenly swept away to a parallel world where Earth is a "Mystic Moon" in the sky. There, she accompanies Van, new king of a kingdom, as he tries to defeat the villains that had attacked and destroyed his country. It's a unique mix of fantastic magic and mecha action and a real classic.
YuYu Hakusho: Yusuke Urameshi was a junior high delinquent with a good heart but a short temper and a powerful fist. Then one day he died trying to save a kid from being hit by a car. With a little work, the pretty Grim Reaper Botan helps him to come back to life, but this revival gives him spiritual awareness and powers, particularly something he calls "spirit gun" which allows him to shoot out a blast of spiritual energy at his enemies. He is then forced into the role of "Spirit Detective", a human tasked with stopping youkai that'd smuggled their way into the human world, which leads to him facing steadily larger and greater threats to mankind and growing more and more powerful in the process.

Accel World: Haruyuki is a short, fat wimp in junior high who's always getting bullied. Using the "Neuro Linker" that provides augmented reality and access to a virtual full-scale virtual net, he tends to escape into online games, where his speed is unbeatable. One day, the most popular girl in school approaches him and invites him to a new "game" called Brain Burst, which accelerates one's brain a thousand times, making everything seem to stop, and contains a special battle system using avatars born from the user's personal traumas and fears.

Kenichi the Mightiest Disciple is a rather fun series as well. It's got some fanservice focused on the main heroine, who tends to sometimes perform combat moves inadvisable when wearing a skirt and has a very impressive chest, but it's really not that bad. Kenichi is a wimp, plain and simple. He's always getting beaten and bullied, and when he joins the Karate club to try and learn self defense, he's just bullied even more there. Finally, his classmate Miu introduces him to the dojo where she lives with her grandfather and five other masters of a broad variety of martial arts. In desperation he agrees to be everyone's disciple. Thus begins an unending barrage of training from six masters who don't know how to hold back, and the only way he survives is with the help of a master whose medical skills can literally raise the dead (and literally does just that any time Kenichi trains with the Muay Thai master). He gradually grows stronger and stronger, and determines to use this strength to protect the weak, but instead of his strength keeping the bullies away, it just draws more and more villains to him. Despite this, he still commits to do whatever it takes until one day he'll be strong enough to protect Miu (rather than struggling to protect enemies from her).
There are also a few series I haven't seen personally but have heard are really good and popular, such as Hunter X Hunter and the old Kenshin series...