The Mahler measure of a polynomial P(x1,x2,...,xn) is the average value of log|P| along the unit n-torus Tn, defined by |xi| = 1 for all i. Interest in this quantity arose from the fact that the Mahler measure of certain polynomials is quite remarkable and not just any random real num- ber – they evaluate to special values of L-functions! However, in general, it is very difficult to evaluate Mahler measures of multivariable polynomi- als. In this talk, we will discuss an invariant property of Mahler measures that lets us resolve some conjectural identities. We will also evaluate the exact Mahler measure of families of polynomials that contain, for every integer n > 1, an n-variable polynomial. We will see how the structure of these polynomials lets us compute their Mahler measures as combinations of values of the Riemann zeta function and values of certain Dirichlet L- functions. This talk includes joint work with Matilde Lal ́ın and Subham Roy.