Given that the polynomial $x^{2}+kx+25$ is a perfect square trinomial, we aim to find the value of $k$.

A perfect square trinomial takes the form $(x+a)^{2} = x^{2} + 2ax + a^{2}$. Comparing this with our given polynomial, we can see that $a^{2} = 25$, which implies $a = \pm 5$. Therefore, our polynomial can be written as:

$$x^{2}+kx+25 = x^{2} + 2ax + a^{2}$$

Given $a = \pm 5$, we substitute $a$ into the equation:

$$x^{2}+kx+25 = x^{2} + 2(\pm 5)x + (\pm 5)^{2}$$

This simplifies to:

$$x^{2}+kx+25 = x^{2} \pm 10x + 25$$

From the equation above, we compare the coefficients of $x$ on both sides to find the value of $k$:

$$kx = \pm 10x$$

Dividing both sides by $x$ (assuming $x \neq 0$), we get:

$$k = \pm 10$$

Therefore, the value of $k$ that makes $x^{2}+kx+25$ a perfect square trinomial is $\boxed{\pm 10}$.