There are 2021 light bulbs in a row, labeled 1 through 2021, each with an on/off switch.  They all start in the off position when 1011 people walk by.  The first person flips the switch on every bulb; the second person flips the switch on every 3rd bulb (bulbs 3, 6, etc.); the third person flips the switch on every 5th bulb; and so on.  In general, the $k$th person flips the switch on every $(2k - 1)$th light bulb, starting with bulb $2k - 1$.  After all 1011 people have gone by, how many light bulbs are on?
1. **Understanding the Problem:**
   - We have 2021 light bulbs initially in the off position.
   - 1011 people walk by, each flipping the switch on certain bulbs.
   - The $k$-th person flips the switch on every $(2k-1)$-th bulb.

2. **Identifying the Pattern:**
   - The first person flips every bulb (1, 2, 3, ..., 2021).
   - The second person flips every 3rd bulb (3, 6, 9, ...).
   - The third person flips every 5th bulb (5, 10, 15, ...).
   - In general, the $k$-th person flips every $(2k-1)$-th bulb.

3. **Determining the Final State of Each Bulb:**
   - A bulb will be on if it is flipped an odd number of times.
   - A bulb $i$ is flipped by the $k$-th person if $i$ is a multiple of $(2k-1)$.
   - Therefore, we need to count how many times each bulb is flipped.

4. **Counting the Flips:**
   - A bulb $i$ is flipped by the $k$-th person if $i$ is a multiple of $(2k-1)$.
   - This means we need to find the number of odd divisors of $i$.

5. **Odd Divisors and Perfect Squares:**
   - A number has an odd number of total divisors if and only if it is a perfect square.
   - This is because divisors generally come in pairs, except for perfect squares which have a middle divisor that is repeated.

6. **Finding Perfect Squares:**
   - We need to count the perfect squares up to 2021.
   - The largest integer $n$ such that $n^2 \leq 2021$ is $n = \lfloor \sqrt{2021} \rfloor = 44$ (since $44^2 = 1936$ and $45^2 = 2025$).

7. **Conclusion:**
   - There are 44 perfect squares between 1 and 2021.
   - Therefore, there are 44 bulbs that will be on after all 1011 people have walked by.

The final answer is $\boxed{44}$.