The (logarithmic) Mahler measure of an $n$-variable Laurent polynomial $P$ is defined by $m(P)=\int_0^1\cdots \int_0^1 \log |P(e^{2\pi i \theta_1},\ldots,e^{2\pi i \theta_n})|\,d\theta_1\cdots d\theta_n.$ In some certain cases, Mahler measures are known to be related to special values of $L$-functions. We will present some new results relating the Mahler measures of polynomials whose zero loci define elliptic curves, $K3$ surfaces, and Calabi-Yau threefold of hypergeometric type to $L$-values of elliptic modular forms. A part of the talk is joint work with Matt Papanikolas and Mat Rogers.