In this paper, the author obtains the boundedness of certain singular integral operators along curves and surfaces with highly singular kernels. In the lower-dimensional case, the author studies the operator \[ Tf(u,v)= \text{p.v. }\int f(u-y, v-\gamma(y)) e^{i|y|^{-\beta}} y^{-1}|y|^{-\alpha} h(y) dy, \] where \(u,v,y\in \mathbb{R}\), \(h\) is a bounded, differentiable and even function, \(\gamma\) is a measurable even function such that \(|\gamma'(r)|\) is increasing on \((\text{supp }\gamma')\cap [0,\infty)\). Assume that either \(h\) is monotone or \(h'\in L^1(\mathbb{R})\) and that \(\gamma'\in L^1(\mathbb{R})\) or \(\gamma\in L^\infty(\mathbb{R})\) and \(\gamma(r)\) is monotone on \([0,\infty)\). The author proves that \(T\) is bounded on \(L^2(\mathbb{R}^2)\) provided \(0 0} r^{-1} \int_{|t|< r}|g(s-\gamma(t))|dt \] is bounded on \(L^p(\mathbb{R}^n)\) for \(1< p<\infty\). The author also extends the above result to the higher-dimensional case, where the singular integral operator \(Tf\) is defined as \[ Tf(x, x_n)= \text{p.v. }\int_{\mathbb{R}^{n-1}} f(x- y,x_n- \gamma(|y|)) e^{i|y|^{-\gamma}} h(y) \Omega(y)|y|^{-n+1-\alpha} dy, \] where \(x,y\in \mathbb{R}^{n-1}\), \(x_n\in \mathbb{R}\), \(\Omega\in L^q(S^{n-2})\) is homogeneous of degree zero and satisfies mean zero over the sphere \(S^{n-2}\). For more details and a weaker condition \(\Omega\in H^1\), the Hardy space, the reader can see another paper by the same author [``Singular integral operators along surfaces of revolution'', J. Math. Anal. Appl. 274, No. 2, 608-625 (2002; Zbl 1022.42008)].