$L$-series of elliptic curves and Mahler measures

For a 2-variate Laurent polynomial $P(x,y)$ the (logarithmic)
Mahler measure is an arithmetic mean of $\log|P|$ on the
torus $|x|=|y|=1$ in $\mathbb C^2$. Famous conjectures due
to Boyd express the Mahler measures of polynomials
$P=x+1/x+y+1/y+c$, $(1+x)(1+y)(x+y)-cxy$, and $x^3+y^3+1-cxy$
in terms of the $L$-series $L(E,2)$ of the elliptic curve $E:P(x,y)=0$.
In my talk I will overview the results and methods of our recent
work with Mat Rogers towards Boyd's conjectural evaluations.