The (logarithmic) Mahler measure of a non-zero rational function P in n variables is defined as the arithmetic mean of log |P| restricted to the standard n-torus (T^n = {(x_1, ..., x_n) ∈ (C*)^n: |x_i| = 1, for all 1 ≤ i ≤ n}) with respect to the unique Haar measure (normalized arc measure) on T^n. It has connections to heights, hyperbolic volumes, arithmetic dynamics, and special values of L-functions. Various generalizations of this definition exist in the literature. This thesis is dedicated to exploring two such generalizations: firstly, when the unit torus is substituted by a torus with arbitrary radii (T_{a_1, ..., a_n})^n = {(x_1, ..., x_n) ∈ (C*)^n: |x_i| = a_i, for all 1 ≤ i ≤ n} (referred to as the \textit{generalized Mahler measure}), and secondly, when the normalized arc measure on the unit torus is replaced by the normalized area measure on the unit disk (referred to as the \textit{areal Mahler measure}). Our primary objective is to quantify the behavior of the Mahler measure of P under such alterations. This thesis is structured into five projects. 1. In Chapter 1, we investigate the definition of the generalized Mahler measure for all Laurent polynomials in n-variables when they do not vanish on the integration torus. This work has been published in [106]. 2. In Chapter 2, we exhibit some nontrivial evaluations of the areal Mahler measure of multivariable polynomials, defined by Pritsker. This is a joint work with Lalin, and has been published in [84]. 3. In Chapter 3, we investigate how the areal Mahler measure changes with the power change of variables. This a joint work with Lalin, and has been published in [83]. 4. In Chapter 4, we investigate the Mahler measure of a particular family of rational functions with an arbitrary number of variables and an arbitrary degree in one of the variables. This is a joint work with Lalin and Nair, and will appear in [81]. 5. In Chapter 5, we evaluate the areal Mahler measure of a family of polynomials using the areal analogue of the Zeta Mahler measure. This is an ongoing joint work with Lalin, Nair, and Ringeling.

In this thesis we consider a variation of the Mahler measure where the defining integral is performed over a more general torus. Our work is based on a tempered family of polynomials originally studied by Boyd, Boyd P_k (x, y) = x + 1/x + y + 1/y + k with k ∈ R_{>4}. For the k = 4 case we use special values of the Bloch-Wigner dilogarithm to obtain the Mahler measure of P_4 over an arbitrary torus (T_ {a, b})^2 = {(x, y) ∈ C* X C* : | x | = a, | y | = b } with a, b ∈ R_{> 0}. Next we establish a relation between the Mahler measure of P_8 over a torus(T_ {a, √a} )^2 and its standard Mahler measure. The combination of this relation with results due to Lalin, Rogers, and Zudilin leads to a formula involving the generalized Mahler measure of this polynomial given in terms of L'(E, 0). In the end, we propose a strategy to prove some similar results for the general case k > 4 over (T_ {a, b})^2 with some restrictions on a, b.