Zeroes may occur in this puzzle, but not as the first digit of a number.
Statements that may help you, but are not really necessary for solving this puzzle are marked with (*).

ACROSS A: V down + Y across
D: D across is a multiple of Y down
I: I across equals the first digit of I across multiplied with W down
J: J across is a multiple of V across
K: a prime number
L: a multiple of U across
M: (*) A' across minus D down minus the digit sum of S down
N: the first digit of N across multiplied with the reverse of R across
P: (*) has the same digit sum as W across
Q: a prime number
R: a prime number
T: has the same digit sum as W down
U: a multiple of the digit sum of B down
V: Divide Y across by the sum of the digit sum of A down and the digit sum of T across, and the result is a divisor of V across.
W: (*) a multiple of the average of the digit sums of K across and T across
X: A palindrome and prime number
Y: a multiple of the sum of A down's and T across's digit sums
Z: a cubic number
A': D down + M across + the digit sum of S down

DOWN A: a multiple of F down
B: I across + 1
C: (*) the digit sum of C down equals Y down
D: the reverse of S down multiplied with V down
E: the fourth power of K across
F: To get F down, multiply G down with T across and subtract this result from E down.
G: (*) The digit sum of G down is a square number.
H: (*) has the same digit sum as S down
J: (*) a palindrome
L: (*) a palindrome
M: a prime number
O: a multiple of the digit sum of G down
R: a multiple of R across
S: a prime number
U: the reverse of the sum of V down and S down
V: multiply all digits of S down. Take this digit product, and again, multiply all of its digits. The result is a divisor of V down.
W: the digit product of Z across
X: the reverse of X across
Y: a palindrome
Z: a square number