For $x\in \mathbb{R} ^n=\mathbb{R} \times\mathbb{R} ^{n-1},$ with $x=(p,y), \, p\in\mathbb{R} ^{n-1}, y\in\mathbb{R} ,$ and certain sets $A\subset \mathbb{R} ^{n-1}$ and functions $u$ on $A,$ we show the existence of harmonic functons $U(x)$ on the upper half-space $\{x=(p,y)\in \mathbb{R} ^n, \, y>0\},$ having prescribed vertical limits $U(p,y)\to u(p),$ as $y\downarrow 0.$ Analogous results are considered for domains in $\mathbb{R} ^n$ consisiting of point $x=(p,y),\, y>L(p),$ lying above the graph $\{(p,y): y=L(p)\}$ of a function $L(p)$ defined on a domain in $\mathbb{R} ^{n-1},$ as well as for polydomains and starlike domains. Such questions are also investigated for holomorphic functions of several variables.