The work «Algorithms with optimal accuracy for the solution of integral equations» is dedicated to construction and investigation of optimal by accuracy numerical methods for the solution of wide class of Fredholm integral equations, namely, ill-posed problems. Among the solved problems are the following: a new projection and piecewise-constant discretization schemes for the solution of wide class of the first kind integral equations are introduced; an approach to adaptive discretization of ill-posed problems is investigated; the accuracy for some fully projection methods in Sobolev space metrics are found both for a posteriori and a priori choice of discretization level; optimal methods for the numerical solution of severely ill-posed problems are proposed; optimal methods of approximate solving for the second-kind Fredholm integral equations with coefficients of Sobolev’s smoothness type are developed and exact order estimates for the information and algorithmic complexity of these methods are established.