  Let $\mathbb{F}_q$ be a finite field with $q=p^n$ elements. In this paper, we
study the number of solutions of equations of the form $a_1 x_1^{d_1}+\dots+a_s
x_s^{d_s}=b$ with $x_i\in\mathbb{F}_{p^{t_i}}$, where $b\in\mathbb{F}_q$ and
$t_i|n$ for all $i=1,\dots,s$. In our main results, we employ results on
quadratic forms to give an explicit formula for the number of solutions of
diagonal equations with restricted solution sets satisfying certain natural
restrictions on the exponents. As a consequence, we present conditions for the
existence of solutions. We also discuss further questions concerning equations
with restricted solution sets and present some open problems.
