Abstract: In this paper, we extend Rothemund and Winfree's examination of the tile complexity of tile self-assembly. They provided a lower bound of$\Omega(\frac{\log N}{\log\log N})$ on the tile complexity of assembling an N x N square for almost all N. Adleman et al. gave a construction which achieves this bound. We consider whether the tile complexity for self-assembly can be reduced through several natural generalizations of the model. One of our results is a tile set of size$O(\sqrt{\log N})$ which assembles an N x N square in a model which allows flexible glue strength between non-equal glues. This result is matched by a lower bound dictated by Kolmogorov complexity. For three other generalizations, we show that the$\Omega(\frac{\log N}{\log\log N})$ lower bound applies to N x N squares. At the same time, we demonstrate that there are some other shapes for which these generalizations allow reduced tile sets. Specifically, for thin rectangles with length N and width k, we provide a tighter lower bound of$\Omega(\frac{N^\frac{1}{k}}{k})$ standard model, yet we also give a construction which achieves$O(\frac{\log N}{\log\log N})$ complexity in a model in which the temperature of the tile system is adjusted during assembly. We also investigate the problem of verifying whether a given tile system uniquely assembles into a given shape, and show that this problem is NP-hard.