What can we say about the configuration space of $n$ points in $\mathbb{R}^{m}$ with fixed distances between the points ?

e.g. for 2 points in $\mathbb{R}^{3}$ with fixed distance between them,the configuration space is $S^{2}$ and for 3 points in $\mathbb{R}^{3}$ with known distances between them, the configuration space is $SO(3)$.

A similar problem is to ask the configuration space of a point in $\mathbb{R}^{m}$ such that its distances from $n$ fixed points is known.

e.g. Consider a point in $\mathbb{R}^{3}$ such that its distances from 2 fixed points is $r_{1}$ and $r_{2}$. Then the point can lie on a sphere of radius $r_{1}$ around point 1 and also on a sphere of radius $r_{2}$ around point 2. Thus the configuration space is the intersection of these 2 sphere, which generically is a circle. (Correct if I'm wrong.)

What is known about these problems in general for any $n,m$ ?