Self-similarity, as the name itself suggests, is a geometric concept which suggests the resemblance of shapes; it has been observed in many disciplines. Scale invariance is a more tangible form of self-similarity where at any magnification there is a smaller piece of the object that is similar to the whole; one may find it is then not difficult to attach this meaning to distribution curves, or surfaces in more general contexts. The bulk of this article under review centers around the anisotropic growth of the sample paths for {\it self-similar} random fields, which may find applications in statistical hydrology and image processing. There are multiple ways to introduce this object of study in higher dimensions. Letting $\Bbb{R}_{+}=[0,\infty)$ and ${\bf t}=(t_1,\dots, t_n)$, a random field $X({\bf t})$ in $\Bbb{R}^n$ is said to be self-similar with index ${\bf H}=(H_1, \dots, H_n)^{\prime}\in \Bbb{R}_{+}^n $ if, for every interior point ${\bf a}\in \Bbb{R}_{+}^n$, $ \{ X({\bf a}\cdot {\bf t}), {\bf t}\in \Bbb{R}^n\} $ has the same distribution as $ \{a_1^{H_1} \cdots a_n^{H_n} X({\bf t}), {\bf t}\in \Bbb{R}^n\} $. For $n=2$, a well-known example is given by the {\it fractional Brownian sheet} $B_{{\bf H}}=\{B_{{\bf H}}(t_1,t_2), t_1\geq 0, t_2\geq 0 \}$, which is a centered Gaussian field with covariance function $$ \Bbb{E}[B_{\bf H}({\bf t}) B_{{\bf H}}({\bf s})]={1\over 4} \prod_{i=1}^2 (|t_i|^{2H_i}+|s_i|^{2H_i}-|t_i-s_i|^{2H_i}), \tag1 $$ where $H_i\in (0,1)$, $i=1,2,$ are called Hurst parameters. To verify its self-similarity with the aid of (1), one can readily check the distributional invariance under the scaling transformation $S_{\bf a}^{\bf H}$, i.e., $(S_{{\bf a}}^{{\bf H}} B_{{\bf H}})({\bf t})= a_1^{-H_1}a_2^{-H_2} B_{{\bf H}}(a_1 t_1, a_2 t_2)$. In §3, the authors' goal is to show the ergodicity of a fractional Brownian sheet; for that purpose they introduce the shift transformation $\theta_{{\bf u}} B_{{\bf H}}({\bf s})=B_{{\bf H}}(s_1+u_1, s_2+u_2)$ and realize $\tau_{{\bf H}} \circ S_{{\bf a}}^{{\bf H}}=\theta_{{\bf u}} \circ \tau_{{\bf H}}$ by actions on $B_{{\bf H}}$, where $\tau_{{\bf H}}$ is the usual Lamperti transformation sending self-similarity to stationarity. It then follows that the ergodicity of the transformed field, now stationary, will imply that of the original field; and for stationary Gaussian field, the ergodicity of the scaling transformation can be adequately studied using the covariance structure. Assuming self-similarity and ergodicity, the authors further investigate certain sample path properties of the random fields; in particular, they introduce the {\it upper} and {\it lower} functions to regularize the asymptotic growth of paths at both the origin and infinity of the first quadrant, and, prove a zero-one law in Theorem 4.1. Section 5 is devoted to strong limit theorems which, again, tell us how quickly the sample paths may grow both at ${\bf 0}$, and $\infty$. To this aim one important random quantity, for a two-dimensional ergodic random field $X$, is given by $$ X^{*}(\omega)=\sup_{0\leq t_1, t_2 \leq 1} |X({\bf t}, \omega)|, \tag2 $$ since its excursion probability will reveal good information about the sample paths on any compact intervals due to self-similarity. It is shown, in Theorem 5.1, that if there is a nondecreasing continuous function $f\: \Bbb{R}_{+}\to (0,+\infty)$ such that $\Bbb{E}[f(X^{*})]$ is finite, and assuming one can find certain {\it well-behaving} $\varphi\: \Bbb{R}_{+}^2 \to (0,+\infty)$ for which $ [xf(\varphi(x_1, x_2))]^{-1} $ is integrable on $ [1,+\infty), $ it then follows that $$ \limsup_{s_1\wedge s_2 \to \infty} \frac{|X({\bf s})|}{s_1^{H_1} s_2^{H_2}\varphi({\bf s})} \tag3 $$ remain universally bounded with probability one. A similar result also holds for the behavior of the random field at ${\bf 0}$. In order to seek applications of (3), one certainly has the desire to choose the right $\varphi$ so that a better bound can be achieved; this has its constraint in how $f$ can be chosen in view of the structure in (2). These issues are addressed in Section 6, along with some concrete examples and explicit calculations; for example, Lemma 6.1 shows that if there is a bound for the tail probability $\Bbb{P}(X^{*}\geq x)\leq a(x)$, say, then in order to see $\Bbb{E}[f(X^{*})]<\infty$ it is sufficient to find positive nondecreasing $f$ for which $f^{\prime}(x)a(x)$ is integrable.