The mathematical representation of complex material behavior requires a sophisticated constitutive formulation, as it is the case of multisurface plasticity. Hence, a complex elastoplastic model demands a robust integration procedure for the plastic evolution equations. Developing integration schemes for plasticity models is an important research topic because these schemes are directly related to the accuracy and efficiency of numerical simulations for materials such as metals, concrete, soils and rocks. The performance of the finite element solution is directly influenced by the convergence characteristics of the state-update procedure. Therefore, this work explores the implementation of complex constitutive models, focusing on generic multisurface plasticity models. This study formulates and evaluates state-update algorithms that form a robust framework for simulating materials governed by multisurface plasticity. Implicit integration algorithms are developed with an emphasis on achieving robustness, comprehensiveness and flexibility to handle cumbersome plasticity applications effectively. The state-update algorithms, based on the backward Euler method and the Newton-Raphson and Newton-Krylov methods, are formulated using line search strategies to improve their convergence characteristics. Additionally, a substepping scheme is implemented to provide further robustness to the state-update procedure. The flexibility of the algorithms is explored, considering various stress conditions such as plane stress and plane strain states, within a single, versatile integration scheme. In this scenario, the robustness and performance of the algorithms are assessed through classical finite element applications. Furthermore, the developed multisurface plasticity background is applied to formulate a coupled elastoplastic-damage model, which is evaluated using experimental tests in concrete structures. The achieved results highlight the effectiveness of the proposed state-update algorithms in integrating multisurface plasticity equations and their ability to handle challenging finite element problems.