Space-time code designs commonly rely on the assumption of independent and identically distributed (i.i.d.) Rayleigh channels. However channel measurements have shown that real-world channels are spatially correlated. In this paper, we study the impact of spatial correlation on the performance of space-time codes evolving in fast Rayleigh fading channels. It is shown that transmit correlation may highly affect the achievable diversity if the code is not properly designed. Therefore, we propose a new design criterion that subsumes the original design criterion developed by Tarokh et al. and leads to space-time codes that are robust in a wide variety of propagation conditions. No channel knowledge is assumed at the transmitter. Codes satisfying this criterion are shown to be much more robust on spatially correlated fast fading channels than codes designed only for i.i.d. fast fading channels. Examples of Space-Time Trellis Codes are proposed in order to illustrate the design criterion.