  We describe how the quadratic Chabauty method may be applied to explicitly
determine the set of rational points on modular curves of genus $g>1$ whose
Jacobians have Mordell--Weil rank $g$. This extends our previous work on the
split Cartan curve of level 13 and allows us to consider modular curves that
may have few known rational points or nontrivial local height contributions at
primes of bad reduction. We illustrate our algorithms with a number of examples
where we determine the set of rational points on several modular curves of
genus 2 and 3: this includes Atkin--Lehner quotients $X_0^+(N)$ of prime level
$N$, the curve $X_{S_4}(13)$, as well as a few other curves relevant to Mazur's
Program B. We also describe the computation of rational points on the genus 6
non-split Cartan modular curve $X_{\textrm{ns}} ^+ (17)$.
