In this talk, I will consider the following geometric connection and separation problems: Given a set of curves and two points in the plane, compute (i) a minimum-size subset of the curves one needs to remove so that there is a path connecting the points that does not intersect any of the remaining curves, and (ii) a minimum-size subset of the curves one needs to retain so that any path connecting the two points intersects some of the retained curves. These problems are motivated by the so-called `resilience' and `1-barrier' properties in sensor networks. While the first one in NP-hard, even for straight-line segments, the second one is polynomial-time solvable. The algorithm uses tools from topological graph theory, such as the 3-path-condition and the fundamental cycle method. Several other generalisations and open problems will also be discussed, while no prior knowledge of computational geometry or topology will be assumed.