The Hausdorff dimensions of the image \(X(t)\), \(t \in E \subset R^ N\), \(N \geq 1\), and the graph \(\text{Gr }X(E) = \{(t,X(t)), t \in E\}\) are found out for the random fields with independent components. The results are extended to self-similar processes and fields (also vector-valued) with stationary increments, including Brownian motion, Brownian sheet, fractional Brownian motion. The Hausdorff dimensions of sample paths are given also for vector-valued processes with strictly stable, \((\alpha, \beta)\)-fractional stable and multi-parameters stable components.