Flat surfaces have two types of symmetries: isometries and affine symmetries. In this talk, we ask “what groups of symmetries are realized by flat structures on a fixed topological surface of infinite type?” Joint with Artigiani, Randecker, Valdez and Weizte-Schmithüsen, we approach these problems using ideas of Aougab, Patel and Vlamis. We culminate with a complete answer to this question for isometries and some encouraging results for affine symmetries.