So-called 4R and 5R loops are mechanisms consisting of 4 or 5 links, pairwise joined by revolute joins. Here, they are supposed to be without offsets, that is, the common perpen-diculars to the joins axes of two neighbour links are concurrent. A complete classification of 4R loops is given, showing, in particular, that all mobile 4R loops were already known. Also, in each case, the assembly conditions on the design parameters have been computed and are explicitly given. For general 5R loops, the assembly condition is a homogeneous polynomial in 10 variable of degree probably higher than 100, too large to be computed, and even too large to be stored in available memory devices. Instead, the configuration ideal of the relations between the design parameters and the position variables has been computed. It is the prime ideal of a projective variety of dimension 8 and degree 1072 in the projective space of dimension 18. This allows us to classify the mobile 5R loops, which appear to be all already known, under a conjecture for which heuristic evidence is provided. These result have been obtained by associating Gröbner basis computations with considerations of classical geometry and algebraic geometry.