To solve the equation $2013^x + 2014^y = 2015^z$ for non-negative integer solutions $(x, y, z)$, we will explore small values manually and check if they satisfy the equation due to the rapid growth of exponential terms.

1. Initial Consideration:
Consider small non-negative integers for $x$, $y$, and $z$ to manually check for solutions.

2. **Checking $x = 0$**:
- When $x = 0$, the equation simplifies to:
$$1 + 2014^y = 2015^z$$
- Consider $y = 1$:
$$1 + 2014 = 2015 = 2015^1$$
This implies $z = 1$.

3. Conclusion:
We found a solution $(x, y, z) = (0, 1, 1)$.

To verify that there are no other solutions, notice that for larger powers, the growth rate of $2015^z$ quickly exceeds that of $2013^x + 2014^y$ unless the base and powers are all particularly small or in balance, which we checked above.

Thus, the only non-negative solution is:
$$\boxed{(0, 1, 1)}$$