An element f∈L1(G),where G is a locally compact group,is called a projection if it is a selfadjoint idempotent(f*f = f* = f).There are several problems which have beenstudied for many years: For which G do there exists nontrivial projections in L1(G)? can one construct explicitly a projection when one exists? Can one describe all projections? We will survey the results so far and spend some time on what is arguably the easiest nonabelian case of two dimensional crystal groups where the questions are already illuminating.