Question: Can the map command work for functions of two variables

For a function of one variable, I use the map command quite a bit to evaluate the function at several values of the independent variable. For example,

f:=x->x^2:

xlist:=[1,2,3]:

map(f,xlist)

How may I do something similar for a function of two variables? For example,

f:=(x,y)->x^2+y^2:

xylist:=[[1,1],[1,2],[2,1],[2,2]]:

Can the map command be used to evaluate f at each ordered pair? Maybe I'm not even using the correct data structure. Thanks!

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