This book describes new forms of layer-damping functions eliminating singularities of solutions to singularly--perturbed problems and corresponding layer-resolving grids -- functions and grids above and beyond those already well known and having broad acceptance, namely, those developed by Bakhvalov and Shishkin. The grids developed by Bakhvalov and Shishkin have been applied to diverse problems, but only to problems with exponential-type layers, typically represented by functions $\exp(-bx/\varepsilon^k)$, occurring in problems for which the solutions of reduced $(\varepsilon=0)$ problems do not have singularities. Such grids are not suitable for tackling other, wider, layers, and also require knowledge of the constant $b$ affecting the width of the exponential layer -- when such knowledge is not always available, for example, for boundary layers in fluid-dynamics problems modeled by Navier-Stokes equations, or for interior layers in solutions to quasilinear nonautonomous problems. One spectacular example of the new layer-resolving grids being presented in the current paper, engendered by a function $\varepsilon^{rk}/(\varepsilon^k +x)^r$, $r>0$, is suitable for dealing not only with exponential layers having arbitrary widths, but with power-type of kind $1$ layers occurring in problems for which the solutions of reduced problems have singularities as well. Other examples of new layer--resolving grids are aimed at dealing with power-type of kind $2$ layers represented by functions $(\varepsilon^k +x)^r$, $0< r < 1$; logarithmic layers represented by functions $\ln(\varepsilon^k +x)/\ln\varepsilon^k $; and mixed layers. It seems that the new layer-resolving grids described in this book should empower researchers to solve broader and more important classes of problems having not only exponential-, but power-, logarithmic-, and mixed-type boundary and interior layers. The book also demonstrates applications of the new layer-resolving grids to numerical solutions for two-point boundary-value problems having diverse types of boundary layers.