We study the problem of memory capacity in balanced networks of spiking neurons. Associative memories are represented by either synfire chains (SFC) or Hebbian cell assemblies (HCA). Both can be embedded in these balanced networks by a proper choice of the architecture of the network. The size WE of a pool in an SFC or of an HCA is limited from below and from above by dynamical considerations. Proper scaling of wE by √K, where K is the total excitatory synaptic connectivity, allows us to obtain a uniform description of our system for any given K. Using combinatorial arguments, we derive an upper limit on memory capacity. The capacity allowed by the dynamics of the system, αC, is measured by simulations. For HCA, we obtain αC of order 0.1, and for SFC, we find values of order 0.065. The capacity can be improved by introducing shadow patterns, inhibitory cell assemblies that are fed by the excitatory assemblies in both memory models. This leads to a doubly balanced network, where, in addition to the usual global balancing of excitation and inhibition, there exists specific balance between the effects of both types of assemblies on the background activity of the network. For each of the memory models and for each network architecture, we obtain an allowed region (phase space) for WE√K in which the model is viable.