We study the Maximum Independent Set (MIS) problem under the notion of stability introduced by Bilu and Linial (2010): a weighted instance of MIS is $γ$-stable if it has a unique optimal solution that remains the unique optimum under multiplicative perturbations of the weights by a factor of at most $γ\geq 1$. The goal then is to efficiently recover the unique optimal solution. In this work, we solve stable instances of MIS on several graphs classes: we solve $\widetilde{O}(Δ/\sqrt{\log Δ})$-stable instances on graphs of maximum degree $Δ$, $(k - 1)$-stable instances on $k$-colorable graphs and $(1 + \varepsilon)$-stable instances on planar graphs. For general graphs, we present a strong lower bound showing that there are no efficient algorithms for $O(n^{\frac{1}{2} - \varepsilon})$-stable instances of MIS, assuming the planted clique conjecture. We also give an algorithm for $(\varepsilon n)$-stable instances. As a by-product of our techniques, we give algorithms and lower bounds for stable instances of Node Multiway Cut. Furthermore, we prove a general result showing that the integrality gap of convex relaxations of several maximization problems reduces dramatically on stable instances. Moreover, we initiate the study of certified algorithms, a notion recently introduced by Makarychev and Makarychev (2018), which is a class of $γ$-approximation algorithms that satisfy one crucial property: the solution returned is optimal for a perturbation of the original instance. We obtain $Δ$-certified algorithms for MIS on graphs of maximum degree $Δ$, and $(1+\varepsilon)$-certified algorithms on planar graphs. Finally, we analyze the algorithm of Berman and Furer (1994) and prove that it is a $\left(\frac{Δ+ 1}{3} + \varepsilon\right)$-certified algorithm for MIS on graphs of maximum degree $Δ$ where all weights are equal to 1.