How often does a diophantine equation have a solution?

Following Bhargava and his collaborators, a current problem of interest in the study of Diophantine equations is to count the number of varieties in families which have a rational point. In this talk I will give an overview of recent works in this area and discuss new results which study families of equations that are parametrised by something other than the usual projective space. In particular, we will show that naive conjectures in the style of the Batyrev--Manin conjecture fail to explain what is happening even when the parameter space is a quadric.