Ataru and Lum
On the outside, it seems that Ataru hates Lum. He avoids her advances, runs when she comes, and tries hard to get dates with any other girl he can. Lum loves Ataru, but he certainly isn't a prize - he's a lecher, he's not very bright and not outstanding in the looks department. He doesn't have much of a redeeming personality either. Both are stubborn and frustrated with the other - they make a perfect couple. The last thing Ataru would ever admit is that he loves Lum, even though Lum -- and everyone else -- knows his feelings.
Lum, Benten, Oyuki, and Ran
These four alien girls were best friends as children. Benten is a fiery, wild personality, which compliments, Oyuki's quiet, reserved style. The only one who doesn't like having adventures with the group is Ran, who is resistant to everything.
Lum and Benten
Lum and Benten are of rival races. So, every year they participate in Setsubun festivities. In Japan, Setsubun is a coming-of-spring festival in which Japanese people throw soybeans about the house and chant "Oni wa soto, fuku wa uchi!" (The devils go out, Fortune/happiness comes in). Now look at this statement and keep in mind that Lum is of the Oni race and Benten is the goddess of Luck... ;) Lum and Benten's version of Setsubun involves fierce battle (that Ataru hates getting caught in the middle of). :D
Lum, Ran, and Rei
Rivalry ahoy! Ran considers herself Lum's nemesis. For every little incident that happened when they were children, Ran holds a vengeful grudge. If that isn't enough, the one Ran has her eye on, the gluttonous Rei, was affianced... to Lum! Lum doesn't like Rei and would gladly let Ran have him, if Rei wasn't mad about Lum. What's an alien to do?
Lum, Ataru, Mendou, Shinobu
Lum wants Ataru. Ataru wants Shinobu. Shinobu wants Mendou. Mendou wants Lum. The possibilities are endless.