We consider closed chains of circles C_1,C_2,\ldots,C_n,C_{n+1}=C_1 such that two neighbouring circles C_i,C_{i+1} intersect or touch each other with A_i being a common point. We formulate conditions such that a polygon with vertices X_i on C_i, and A_i on the (extended) side X_iX_{i+1}, is closed for every position of the starting point X_1 on C_1. Similar results apply to open chains of circles. It turns out that the intersection of the sides X_iX_{i+1} and X_jX_{j+1} of the polygon lies on a circle C_{ij} through A_i and A_j with the property that C_{ij}, C_{jk} and C_{ki} pass through a common point. The six circles theorem of Miquel and Steiner's quadrilateral Theorem appear as special cases of the general results.
