Find all pairs of $a$, $b$ of positive integers satisfying the equation $2a^2 = 3b^3$.
1. Start with the given equation:
   \[
   2a^2 = 3b^3
   \]

2. Since \(2a^2 = 3b^3\), we can infer that \(b^3\) must be even (because it is multiplied by 3 to give an even number). Therefore, \(b\) must be even. Let \(b = 2b_0\) for some positive integer \(b_0\).

3. Substitute \(b = 2b_0\) into the equation:
   \[
   2a^2 = 3(2b_0)^3 = 3 \cdot 8b_0^3 = 24b_0^3
   \]
   Simplify to get:
   \[
   a^2 = 12b_0^3
   \]

4. Since \(a^2 = 12b_0^3\), \(a^2\) must be divisible by 3. Therefore, \(a\) must be divisible by 3. Let \(a = 3a_0\) for some positive integer \(a_0\).

5. Substitute \(a = 3a_0\) into the equation:
   \[
   (3a_0)^2 = 12b_0^3 \implies 9a_0^2 = 12b_0^3
   \]
   Simplify to get:
   \[
   3a_0^2 = 4b_0^3
   \]

6. Since \(3a_0^2 = 4b_0^3\), \(b_0^3\) must be divisible by 3. Therefore, \(b_0\) must be divisible by 3. Let \(b_0 = 3b_1\) for some positive integer \(b_1\).

7. Substitute \(b_0 = 3b_1\) into the equation:
   \[
   3a_0^2 = 4(3b_1)^3 = 4 \cdot 27b_1^3 = 108b_1^3
   \]
   Simplify to get:
   \[
   a_0^2 = 36b_1^3
   \]

8. Since \(a_0^2 = 36b_1^3\), \(a_0^2\) must be divisible by 4. Therefore, \(a_0\) must be divisible by 2. Let \(a_0 = 2a_1\) for some positive integer \(a_1\).

9. Substitute \(a_0 = 2a_1\) into the equation:
   \[
   (2a_1)^2 = 36b_1^3 \implies 4a_1^2 = 36b_1^3
   \]
   Simplify to get:
   \[
   a_1^2 = 9b_1^3
   \]

10. Since \(a_1^2 = 9b_1^3\), \(a_1^2\) must be divisible by 3. Therefore, \(a_1\) must be divisible by 3. Let \(a_1 = 3a_2\) for some positive integer \(a_2\).

11. Substitute \(a_1 = 3a_2\) into the equation:
    \[
    (3a_2)^2 = 9b_1^3 \implies 9a_2^2 = 9b_1^3
    \]
    Simplify to get:
    \[
    a_2^2 = b_1^3
    \]

12. Since \(a_2^2 = b_1^3\), we can let \(a_2 = k^3\) and \(b_1 = k^2\) for some positive integer \(k\).

13. Substitute \(a_2 = k^3\) and \(b_1 = k^2\) back to find \(a\) and \(b\):
    \[
    a_0 = 2a_1 = 2(3a_2) = 6k^3
    \]
    \[
    b_0 = 3b_1 = 3(k^2) = 3k^2
    \]
    \[
    a = 3a_0 = 3(6k^3) = 18k^3
    \]
    \[
    b = 2b_0 = 2(3k^2) = 6k^2
    \]

Therefore, the pairs \((a, b)\) that satisfy the equation \(2a^2 = 3b^3\) are given by:
\[
\boxed{(a, b) = (18k^3, 6k^2) \ \text{where} \  k \in \mathbb{N}}
\]