Let $M$ and $N$ be topological spaces, are there necessary and sufficient conditions on the topological properties of the spaces such that $C(M,N)$ is metrizable?

For $M$ compact and $N$ a metric space, the space is obviously metrizable using the uniform convergence topology, $d(f,g)=\sup_{x\in M}d(f(x),g(x))$.

And and also, if $N$ is a metric space, but $M$ is not necessarily compact the space of continuous bounded functions $C_0(M,N)=\{f\in C(M,N)\mid d(f(x),a)\leq K_f, \forall x\in M\}$ for a point $a\in N$ and $K_f>0$ is metrizable with the same distance.

But in general, which distances are usable in $C(M,N)$ in the context of noncompact space $M$.

Let $M$ and $N$ be topological spaces. Are there necessary and sufficient conditions on the topological properties of the spaces such that $C(M,N)$ is metrizable?

For $M$ compact and $N$ a metric space, the space is obviously metrizable using the uniform convergence topology, $d(f,g)=\sup_{x\in M}d(f(x),g(x))$.

And also, if $N$ is a metric space, but $M$ is not necessarily compact the space of continuous bounded functions $C_0(M,N)=\{f\in C(M,N)\mid d(f(x),a)\leq K_f, \forall x\in M\}$ for a point $a\in N$ and $K_f>0$ is metrizable with the same distance.

But in general, which distances are usable in $C(M,N)$ in the context of a noncompact space $M$?