We study the critical behavior of the component sizes for the configuration model when the tail of the degree distribution of a randomly chosen vertex is a regularly-varying function with exponent $τ-1$, where $τ\in (3,4)$. The component sizes are shown to be of the order $n^{(τ-2)/(τ-1)}L(n)^{-1}$ for some slowly-varying function $L(\cdot)$. We show that the re-scaled ordered component sizes converge in distribution to the ordered excursions of a thinned Lévy process. This proves that the scaling limits for the component sizes for these heavy-tailed configuration models are in a different universality class compared to the Erdős-Rényi random graphs. Also the joint re-scaled vector of ordered component sizes and their surplus edges is shown to have a distributional limit under a strong topology. Our proof resolves a conjecture by Joseph, Ann. Appl. Probab. (2014) about the scaling limits of uniform simple graphs with i.i.d degrees in the critical window, and sheds light on the relation between the scaling limits obtained by Joseph and in this paper, which appear to be quite different. Further, we use percolation to study the evolution of the component sizes and the surplus edges within the critical scaling window, which is shown to converge in finite dimension to the augmented multiplicative coalescent process introduced by Bhamidi et. al., Probab. Theory Related Fields (2014). The main results of this paper are proved under rather general assumptions on the vertex degrees. We also discuss how these assumptions are satisfied by some of the frameworks that have been studied previously.