Should I stay with my boyfriend?

We have been dating for a little over a year. In that time he has cheated three times, that I know of. He lies about pretty much everything. He used a social site to find girls. When I found out he deleted it. At nine or ten months he said he would change, but I still find out that he lies. Before we met he would get in relationship and still talk to his ex girlfriends, but he told me he would stop. He has stopped talking to all but one. That one he says he don’t talk to, she just won’t stop texting or calling him. I have seen some of the texts, she talks about him coming over to her house all week then to my house on the weekend. He promises he hasn’t and will not talk to her or see her. Then I find a recording on his phone of them talking back in December. Now I’m pretty sure he created a a social site and has just been lying about not having one. I do love him but with him lying my heart is wanting to run. I don’t know what to do. I try not to let things bother me so much and give him space, but I can’t handle the lying and talking to other girls.