This paper devises a general modeling and analyzing framework for a heterogeneous wireless network (HetNet) in which several wireless subnetworks coexist and use multiple radio access technologies (multi-RATs). The coexisting coverage and network capacity in such a multi-RAT HetNet are hardly investigated in prior works. To characterize the coexisting interactions in a multi-RAT HetNet, in this paper, we consider a HetNet consisting of <inline-formula> <tex-math notation="LaTeX">K </tex-math></inline-formula> tiers of access points (APs) and two different RATs, RAT-<inline-formula> <tex-math notation="LaTeX"> \mathsf {L} </tex-math></inline-formula> and RAT-<inline-formula> <tex-math notation="LaTeX"> \mathsf {U} </tex-math></inline-formula>, are adopted in the HetNet. RAT-<inline-formula> <tex-math notation="LaTeX"> \mathsf {L} </tex-math></inline-formula> is adopted by the APs in the first <inline-formula> <tex-math notation="LaTeX">K-1 </tex-math></inline-formula> tiers and APs in the <inline-formula> <tex-math notation="LaTeX">K </tex-math></inline-formula>th tier only use RAT-<inline-formula> <tex-math notation="LaTeX"> \mathsf {U} </tex-math></inline-formula>. Both noncrossing-RAT and crossing-RAT user association scenarios are considered. In each scenario, the void probability and the channel access probability of the APs in each tier are first found and then the tight lower bounds and their lowest limits on the proposed coexisting coverage and network capacity are derived. We show that multi-RAT networks in general can achieve higher link coverage and capacity by using opportunistic carrier sense multiple access with collision avoidance that avoids/alleviates severe interfering between all coexisting APs. Also, crossing-RAT user association is shown to achieve much higher coexisting coverage and network capacity than noncrossing-RAT user association. Finally, numerical simulations for the LTE-U and WiFi networks coexisting in the HetNet validate our findings.