Find all polynomials $P(x)$ with integral coefficients whose values at points $x = 1, 2, . . . , 2021$ are numbers $1, 2, . . . , 2021$ in some order.
1. **Understanding the Problem:**
   We need to find all polynomials \( P(x) \) with integer coefficients such that the values of \( P(x) \) at \( x = 1, 2, \ldots, 2021 \) are exactly the numbers \( 1, 2, \ldots, 2021 \) in some order.

2. **Using Polynomial Properties:**
   We use the property that for any integers \( a \) and \( b \), if \( P(a) \) and \( P(b) \) are integers, then \( a - b \) divides \( P(a) - P(b) \). This is a consequence of the fact that \( P(x) \) has integer coefficients.

3. **Applying the Property:**
   For \( a, b \in \{1, 2, \ldots, 2021\} \), we have:
   \[
   a - b \mid P(a) - P(b)
   \]
   Since \( P(a) \) and \( P(b) \) are distinct integers from the set \( \{1, 2, \ldots, 2021\} \), \( P(a) - P(b) \) is a non-zero integer. Therefore, \( a - b \) must divide a non-zero integer, implying \( a \neq b \).

4. **Considering the Polynomial Degree:**
   Since \( P(x) \) maps \( \{1, 2, \ldots, 2021\} \) to itself in some order, \( P(x) \) must be a bijection on this set. The polynomial \( P(x) \) must be of degree at least 2021 because it has 2021 distinct roots.

5. **Form of the Polynomial:**
   We consider the polynomial \( Q(x) = P(x) - x \). Since \( P(x) \) is a bijection, \( Q(x) \) must be zero at \( x = 1, 2, \ldots, 2021 \). Therefore, \( Q(x) \) can be written as:
   \[
   Q(x) = (x-1)(x-2)\cdots(x-2021)R(x)
   \]
   where \( R(x) \) is some polynomial with integer coefficients.

6. **Constructing \( P(x) \):**
   Thus, we have:
   \[
   P(x) = x + (x-1)(x-2)\cdots(x-2021)R(x)
   \]
   Similarly, considering the reverse order, we can also have:
   \[
   P(x) = 2022 - x + (x-1)(x-2)\cdots(x-2021)S(x)
   \]
   where \( S(x) \) is another polynomial with integer coefficients.

7. **Conclusion:**
   Therefore, the polynomials \( P(x) \) that satisfy the given conditions are of the form:
   \[
   P(x) = x + (x-1)(x-2)\cdots(x-2021)R(x)
   \]
   or
   \[
   P(x) = 2022 - x + (x-1)(x-2)\cdots(x-2021)S(x)
   \]
   where \( R(x) \) and \( S(x) \) are polynomials with integer coefficients.

The final answer is \( \boxed{ P(x) = x + (x-1)(x-2)\cdots(x-2021)R(x) } \) or \( P(x) = 2022 - x + (x-1)(x-2)\cdots(x-2021)S(x) \).