Let $f(x,y)$ be a polynomial with real coefficients in the real variables $x$ and $y$ defined over the entire $xy$-plane. What are the possibilities for the range of $f(x,y)?$
1. **Continuity and Connectedness of the Range:**
   Since \( f(x,y) \) is a polynomial with real coefficients in the real variables \( x \) and \( y \), it is continuous over the entire \( xy \)-plane. The range of a continuous function over a connected domain (which the \( xy \)-plane is) must be a connected set. In the real numbers, connected sets are intervals.

2. **Unbounded Intervals:**
   We need to show that all unbounded intervals are possible ranges for \( f(x,y) \).
   - For \( f(x,y) = x \), the range is \( \mathbb{R} \).
   - For \( f(x,y) = a + x^2 + y^2 \), the range is \( [a, \infty) \).
   - For \( f(x,y) = a + x^2 + (xy - 1)^2 \), the range is \( (a, \infty) \).

   Similarly, by considering negative signs or other transformations, we can achieve ranges \( (-\infty, a] \) and \( (-\infty, a) \).

3. **Bounded Polynomials:**
   Next, we consider the case where \( f(x,y) \) is bounded. We need to show that if \( f(x,y) \) is bounded, then it must be constant.
   - For a fixed \( x \), consider the function \( g(y) = f(x, x+y) \). Since \( f \) is bounded, \( g(y) \) is a bounded polynomial in one variable \( y \). By the properties of polynomials, a bounded polynomial in one variable must be constant. Therefore, \( f(x, x+y) = c(x) \) for some constant \( c(x) \).
   - This implies \( f(x,y) = f(x,x) \) for all \( x \) and \( y \).
   - Now, consider the function \( h(x) = f(x,x) \). Since \( f \) is bounded, \( h(x) \) is a bounded polynomial in one variable \( x \). Again, by the properties of polynomials, a bounded polynomial in one variable must be constant. Therefore, \( f(x,x) = c \) for some constant \( c \).

   Hence, \( f(x,y) \) is constant, and the only other possibility for its range is a single point \( \{ a \} \).

Conclusion:
The range of \( f(x,y) \) can be either \( \mathbb{R} \), \( [a, \infty) \), \( (-\infty, a] \), \( (a, \infty) \), \( (-\infty, a) \), or \( \{ a \} \) for arbitrary \( a \in \mathbb{R} \).

The final answer is \( \boxed{ \mathbb{R} } \), \( [a, \infty) \), \( (-\infty, a] \), \( (a, \infty) \), \( (-\infty, a) \), or \( \{ a \} \) for arbitrary \( a \in \mathbb{R} \).