Given the equations:

$$a^2 + b = c^2,$$

$$b^2 + c = a^2,$$

$$c^2 + a = b^2,$$

we are tasked with finding all possible values of $abc$.

### Step 1: Analyze the System of Equations

Let's add all three equations:

$$(a^2 + b) + (b^2 + c) + (c^2 + a) = c^2 + a^2 + b^2.$$

Simplifying the left-hand side, we have:

$$a^2 + b + b^2 + c + c^2 + a.$$

This gives us:

$$a^2 + b^2 + c^2 + a + b + c.$$

Thus the equation simplifies to:

$$a^2 + b^2 + c^2 + a + b + c = a^2 + b^2 + c^2.$$

This simplifies further to:

$$a + b + c = 0.$$

### Step 2: Substitution and Solving

Using the relation $a + b + c = 0$, solve for one of the variables, for instance:

$$c = -a - b.$$

Substitute $c = -a - b$ into the original equations to check consistency:

1. From $a^2 + b = c^2$:

$$a^2 + b = (-a-b)^2 = a^2 + 2ab + b^2.$$

Simplifying, we get:

$$b = 2ab + b^2.$$

2. From $b^2 + c = a^2$:

$$b^2 - a - b = a^2.$$

3. From $c^2 + a = b^2$:

$$a^2 + 2ab + b^2 + a = b^2.$$

### Step 3: Simplifying the System

Given the symmetry of the equations and substituting $a + b + c = 0$, this suggests that one or more of $a, b, c$ could be zero. Suppose any of $a, b,$ or $c$ is zero; without loss of generality, let's assume $b = 0$. Then, we quickly check:

1. $a^2 = c^2$ gives $a = \pm c$.

2. $b^2 + c = a^2$ gives $c = a^2$.

3. $c^2 + a = 0$.

These result in contradictions unless similarly $a = 0$ or $c = 0$. Therefore, the only valid solution is that $abc = 0$ when at least one of the variables should be zero.

### Conclusion

From steps outlined above, the only possible value of $abc$ that satisfies the conditions is:

$$\boxed{0}.$$

This solution method verifies that assuming any of the variables to zero holds under the condition $a + b + c = 0$, indicating the necessity of one of $a, b, c$ being zero to satisfy the original equations.