Different approaches to integrability of partial differential equations (pdes) are based on their diverse but related properties such as existence of infinite hierarchies of (local or nonlocal) symmetries and/or conservation laws, zero-curvature representations, Lax integrability, recursion operators etc. This thesis consists of papers: [I] Baran, H. Infinitely many commuting nonlocal symmetries for modified Martinez Alonso–Shabat equation. Communications in Nonlinear Science and Numerical Simulation 96 (2021), 105692. [II] Baran, H., Krasil’shchik, I.S., Morozov, O.I., and Vojčák, P. Nonlocal Symmetries of Integrable Linearly Degenerate Equations: A Comparative Study. Theoretical and Mathematical Physics 196 (2)(2018), 1089–1110. [III] Baran, H., Krasil’shchik, I. S., Morozov, O.I., and Vojčák, P. Coverings over Lax integrable equations and their nonlocal symmetries. Theoretical and Mathematical Physics 188 (3)(2016), 1273–1295. [IV] Baran, H., Krasil’shchik, I.S., Morozov, O.I., and Vojčák, P. Integrability properties of some equations obtained by symmetry reductions. Journal of Nonlinear Mathematical Physics 22 (2)(2015), 210–232. [V] Baran, H., Krasil’shchik, I.S., Morozov, O.I., and Vojčák, P. Symmetry reductions and exact solutions of Lax integrable 3-dimensional systems. Journal of Nonlinear Mathematical Physics 21 (4)(2014), 643–671. [VI] Baran, H. and Marvan, M. Classification of integrable Weingarten surfaces possessing an sl(2)-valued zero curvature representation. Nonlinearity 23 (10) (2010), 2577–2597. [VII] Baran,H. and Marvan, M. On integrability of Weingarten surfaces: A forgotten class. Journal of PhysicsA: Mathematical and Theoretical 42 (40) (2009), 404007. All of them study integrability properties of some or several nonlinear PDE.