Large integral points on elliptic curves

We describe several methods which permit one to search for big integral points on certain elliptic curves, i.e., for integral solutions (x, y) of certain Diophantine equations of the form ${y^2} = {x^3} + ax + b\;(a,b \in {\mathbf{Z}})$ in a large range $\vert x\vert,\vert y\vert \leqslant B$, in time polynomial in $\log \log B$. We also give a number of individual examples and of parametric families of examples of specific elliptic curves having a relatively large integral point.