Here we will introduce Mahler measure of Polynomials (and/or algebraic numbers) and Lehmer's Problem (known as Lehmer Conjecture). Then we will see some recent results toward lehmer problem and equivelent forms of lehmer problem, specially lehmer problem in terms of (logarithmic Weil height) heights of algebraic numbers. Also we will show an improved result of F. Amoroso (University of Torino, Italy) and R. Dvornicich (University of Pisa, Italy), of abelian height. Finally some interesting directions towards future work.