Let $P(x_1,x_2,...,x_n)=0$ be a given polynomial Diophantine equation in $n$ variables with integer coefficients (for example $x_1^2+3x_2-10+x_1x_2^4=0$).
Suppose further that this equation has a solution modulo every natural number $m$. That is for every natural number $m$ we can assign integer values to $x_1,x_2,...,x_n$ such that $P(x_1,x_2,...,x_n)$ is divisible by $m$.
Does this imply that the equation $P(x_1,x_2,...,x_n)=0$ has integer solutions?
(Clearly having a solution modulo every natural $m$ is a necessary condition for $P(x_1,x_2,...,x_n)=0$ having solutions. I'm interseted if it is also a sufficient one.)
As an extreme example, use $x^2+y^2+z^2+w^2+1=0$. There are solutions modulo every $m$, but no integer solutions.
It is useful to add the condition that there are real solutions. There is a large literature. For details, look under Hasse Principle.