  We construct moduli spaces of complex affine and dilation surfaces. Using
ideas of Veech, we show that the the moduli space of affine surfaces with fixed
genus and with cone points of fixed complex order is a holomorphic affine
bundle over the moduli space of Riemann surfaces. Similarly, the moduli space
of dilation surfaces is a covering space of the moduli space of Riemann
surfaces. We classify the connected components of the moduli space of dilation
surfaces and show that any component is an orbifold K(G,1) where G is the
framed mapping class group of Calderon-Salter.
