Mock theta functions were first introduced by Ramanujan. Historically, mock theta functions can be represented as Eulerian forms, Appell-Lerch sums, Hecke-type double sums, and Fourier coefficients of meromorphic Jacobi forms. In this paper, in view of the $q$-Zeilberger algorithm and the Watson--Whipple transformation formula, we establish five three-parameter mock theta functions in Eulerian forms, and express them by Appell--Lerch sums. Especially, the main results generalize some two-parameter mock theta functions. For example, setting $(m,q,x)\rightarrow (1,q^{1/2},xq^{-1/2})$ in $$\sum_{n=0}^{\infty}\frac{(-q^2;q^2)_{n}q^{n^2+(2m-1)n}}{(xq^{m},x^{-1}q^{m};q^2)_{n+1}},$$ we derive the universal mock theta function $g_2(x,q)$.