The study of attractors within random dynamical systems can be approached from both topological and ergodic perspectives. The concept of an attractor is defined in various ways, depending on the viewpoint adopted. Here, we focus primarily on random dynamical systems influenced by deterministic external forces. In discrete time, these systems are often modeled as skew products. A thorough understanding of invariant graphs in skew products is well-established for cases where the fiber is one-dimensional, such as a circle or interval. However, knowledge is limited regarding invariant graphs in scenarios where the fibers are higher-dimensional. Furthermore, we examine skew products in which the fiber maps do not meet the conditions for uniform hyperbolicity. In these cases, the assumption of uniformly contracting fiber dynamics is replaced by a non-uniform contraction condition, which can be described using Lyapunov exponents or partial hyperbolicity. Analyzing and predicting behaviors under these more relaxed conditions, especially by quantifying the stability of invariant graphs with Lyapunov exponents, is still an underdeveloped area

Keywords