Let $a, b, c$ be three non-zero integers. It is known that the sums $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ and $\frac{b}{a}+\frac{c}{b}+\frac{a}{c}$ are integers. Find these sums.
1. **Given Information and Initial Setup:**
   We are given that \(a, b, c\) are non-zero integers and the sums \(\frac{a}{b} + \frac{b}{c} + \frac{c}{a}\) and \(\frac{b}{a} + \frac{c}{b} + \frac{a}{c}\) are integers. We need to find these sums.

2. **Polynomial Construction:**
   Consider the monic cubic polynomial \(f(x)\) with roots \(\frac{a}{b}, \frac{b}{c}, \frac{c}{a}\). By Vieta's formulas, the sum of the roots taken one at a time, the sum of the products of the roots taken two at a time, and the product of the roots are related to the coefficients of the polynomial.

3. **Sum of the Roots:**
   The sum of the roots of \(f(x)\) is given by:
   \[
   \frac{a}{b} + \frac{b}{c} + \frac{c}{a}
   \]
   This is given to be an integer.

4. **Sum of the Reciprocals of the Roots:**
   The sum of the reciprocals of the roots is:
   \[
   \frac{b}{a} + \frac{c}{b} + \frac{a}{c}
   \]
   This is also given to be an integer.

5. **Product of the Roots:**
   The product of the roots is:
   \[
   \frac{a}{b} \cdot \frac{b}{c} \cdot \frac{c}{a} = 1
   \]
   This is clearly an integer (specifically, 1).

6. **Application of the Rational Root Theorem:**
   By the Rational Root Theorem, since \(\frac{a}{b}, \frac{b}{c}, \frac{c}{a}\) are roots of a polynomial with integer coefficients, these roots must be rational numbers. Given that their product is 1, the only possible values for these roots are \(\pm 1\).

7. **Possible Values for the Roots:**
   Since \(\frac{a}{b}, \frac{b}{c}, \frac{c}{a} \in \{-1, 1\}\), we have:
   \[
   \left|\frac{a}{b}\right| = \left|\frac{b}{c}\right| = \left|\frac{c}{a}\right| = 1
   \]
   This implies \(|a| = |b| = |c|\).

8. **Possible Sums:**
   The possible values for \(\frac{a}{b}, \frac{b}{c}, \frac{c}{a}\) are either all 1 or all -1. Therefore, the sums are:
   \[
   \frac{a}{b} + \frac{b}{c} + \frac{c}{a} = 1 + 1 + 1 = 3
   \]
   or
   \[
   \frac{a}{b} + \frac{b}{c} + \frac{c}{a} = -1 + -1 + -1 = -3
   \]

9. **Conclusion:**
   The sums \(\frac{a}{b} + \frac{b}{c} + \frac{c}{a}\) and \(\frac{b}{a} + \frac{c}{b} + \frac{a}{c}\) are both either 3 or -3.

The final answer is \(\boxed{3 \text{ or } -3}\).