There are $n$ distinct lines in three-dimensional space such that no two lines are parallel and no three lines meet at one point. What is the maximal possible number of planes determined by these $n$ lines?

We say that a plane is determined if it contains at least two of the lines.
1. **Understanding the Problem:**
   We need to determine the maximum number of planes that can be formed by \( n \) distinct lines in three-dimensional space, given that no two lines are parallel and no three lines meet at a single point. A plane is determined if it contains at least two of the lines.

2. **Basic Combinatorial Insight:**
   To form a plane, we need at least two lines. Therefore, the problem reduces to counting the number of ways to choose 2 lines out of \( n \) lines.

3. **Using Combinatorial Formula:**
   The number of ways to choose 2 lines out of \( n \) lines is given by the binomial coefficient:
   \[
   \binom{n}{2} = \frac{n(n-1)}{2}
   \]

4. **Verification of Conditions:**
   - No two lines are parallel: This ensures that any pair of lines will determine a unique plane.
   - No three lines meet at a single point: This ensures that each pair of lines will determine a distinct plane without any ambiguity.

5. **Conclusion:**
   Given the conditions, every pair of lines will determine a unique plane. Therefore, the maximal number of planes determined by \( n \) lines is:
   \[
   \binom{n}{2} = \frac{n(n-1)}{2}
   \]

The final answer is \(\boxed{\frac{n(n-1)}{2}}\)