given a smooth projective family $f : x \to s$ defined over the ring of integers of a number field, the shafarevich problem is to describe those fibres of f which have everywhere good reduction. this can be interpreted as asking for the dimension of the zariski closure of the set of integral points of $s$, and is ultimately a difficult diophantine question about which little is known as soon as the dimension of $s$ is at least 2. recently, lawrence and venkatesh gave a general strategy for addressing such problems which requires as input lower bounds on the monodromy of f over essentially arbitrary closed subvarieties of $s$. in this talk we review their ideas, and describe recent work which gives a fully effective method for computing these lower bounds. this gives a fully effective strategy for solving shafarevich-type problems for essentially arbitrary families $f$.