This book develops the theory of quasi-Banach $K_σ$-spaces $\mathbf{F}_ψ(Ω)$, $\mathbf{F}_ψ^*(Ω)$, and $D_{V,W}(Ω)$ of random variables and stochastic processes, extending the classical framework of Orlicz spaces, $Sub_\varphi(Ω)$ and $V(\varphi,ψ)$ spaces. The book consists of eleven chapters. The first two chapters establish the foundational theory: stochastic processes from quasi-Banach $K_σ$-spaces are introduced, and the fundamental properties of $\mathbf{F}_ψ(Ω)$ are studied in detail. The third chapter derives distribution estimates for suprema of processes from $\mathbf{F}_ψ^*(Ω)$, and the fourth addresses approximation theory in $SF_ψ(Ω)$. The fifth chapter examines Orlicz spaces and their connections to $\mathbf{F}_ψ(Ω)$. Chapters six and seven treat the pre-Banach $K_σ$-spaces $D_{V,W}(Ω)$, establishing their essential properties and evaluating reliability and accuracy of stochastic process models. The eighth chapter provides norm distribution estimates in $L_p(T)$ for processes from $\mathbf{F}_ψ(Ω)$. The ninth chapter develops the Monte Carlo method for multiple integrals over $\mathbb{R}^n$ with prescribed reliability and accuracy. The final two chapters treat modeling of $Sub_\varphi(Ω)$ processes - subclasses of $K_σ$-spaces - with given reliability and accuracy in $L_p(T)$ and $C(T)$ respectively. The results are substantially based on the authors' original work and that of their co-authors.