General domain Hankel and Toeplitz operators is a class of operators that significantly extends the classical counterparts to several variables. I will discuss various results concerning their structure, positive semidefinitness and finite rank. It turns out that their symbols are then sums of exponential functions, and these operators therefore have a potential for playing a key role in multidimensional frequency estimation / approximation by sparse exponential sums. I will elaborate on this connection and show some numerical results. Potential applications range from seismic imaging to chemistry (NMR) and medicine (e.g. MRI).