We study the multidimensional aggregation equations with power-law kernels $K$. We prove that with biological relevant potential $K(x)=jxj$, the equation is ill-posed in the critical Lebesgue space $L_{d/(d-1)}(R^d)$. We then extend this result to more general power-law kernels $K(x) = jxj^\alpha$, 0<\alpha<2$ for $p=p_s:= d/(d+\alpha-2)$, and prove a conjecture of Bertozzi, Laurent and Rosado about an instantaneous-mass-concentration phenomenon. Finally, we classify all the ``first kind'' radially symmetric similarity solutions in dimension greater than two.