Viruses of importance to human health that have bats as their reservoir host include: 

Rabies virus (a Rhabdovirus) - Chapter 20

Severe Acute Respiratory Syndrome Coronavirus 1 (SARS-CoV-1), the causative agent of SARS - Chapter 25

Middle East Respiratory Syndrome Coronavirus (MERS-CoV), the causative agent of MERS - Chapter 25

Severe Acute Respiratory Syndrome Coronavirus 2 (SARS-CoV-2), the causative agent of COVID-19 - Chapter 25

Ebola virus (a Filovirus) - Chapter 22

Marburg virus (a Filovirus)
This is named after a town in Germany where 31 workers at an industrial plant caught a form of viral hemorrhagic fever. Seven of the infected people died and so the virus is considered to be extremely dangerous.

Lassa fever virus (an Arenavirus)
This is a hemorrhagic virus that is It is endemic in West African countries, where there are upwards of half a million cases per year leading to 5000 deaths.

Nipah virus (a Henipahvirus), the causative agent of acute encephalitis syndrome - Chapter 22

Hendra virus (a Henipahvirus formerly called "Equine morbillivirus").
An infection of Hendra virus broke out in pigs in 1998 and spread to humans leading to a number of deaths. Hendra virus infection of humans may occur in the lungs resulting in hemorrhage and edema. It can also lead to meningitis in humans. In 2004, there was an outbreak in Australia in horses that in a few cases spread to humans, two of whom died. In horses, the virus usually causes pulmonary edema and congestion and there may be neurological symptoms. This virus has a very high case fatality rate that is 60% in humans and 75% in horses.

All of these human viral diseases have one thing in common, their reservoir host is bats. For Ebola, Nipah and Hendra viruses, the reservoir hosts are various species of fruit bat/flying fox (Pteropus -suborder Yinpterochiroptera) while the others are carried by smaller bats. Bats also carry rubula, morbilli and hepadnaviruses related to human mumps, measles and hepatitis B virus and may have been donors of these viruses to humans although now human to human is the major form of transmission. 

These diseases also have in common that they are caused by single-stranded viruses (except hepadnaviridae which are DNA viruses with an RNA intermediate). They are highly pathogenic and elicit severe symptoms in humans, frequently as a result of aberrant innate immune activation. This dysregulated host immune response is an important contributor to tissue damage and pathogenicity resulting from human infection where there is considerable mortality.

The association with single-strand RNA viruses rather than DNA viruses may reflect the greater mutation rate of the former so that they can skip from their reservoir host to another animal species. 

Thus, in their secondary hosts, these viruses elicit a strong, long lasting, often pathogenic immune response whereas bats show no obvious symptoms, sustain a long term persistent infection and exhibit no or minimal signs of disease. So why are bats so tolerant of viruses and why do they not show pathological effects of these viruses that are so deadly to other animals? The answer is that we do not know because we know little about bat immunology. However, there are some clues and they may involve the lifestyle of bats as the only flying mammals. Bats have a very high metabolic rate and usually a high metabolic rate inversely correlates with longevity but bats live for a comparatively long time. In comparison with other mammals, bats would be expected to live for about four years but some have been found to live nearly 10 times longer than that.  Bats are among only 19 species of mammals that live longer than humans when their body size is taken into account. There are suggestions that this long lifespan derives from greater protein protection from oxidative stress and better repair of DNA damaged as a result of the high metabolic rate that comes from flying.  

But these aspects of bat physiology did not evolve to allow bats to tolerate more viral species asymptomatically than other animals. Nevertheless, these mechanisms may allow bats to lessen viremia (viral load) and there has recently been some information as to how this happens.  

Animals respond to viral infection, that is the presence of viral DNA or RNA, by making type I interferons (interferon alpha and interferon beta). A virus-infected cell releases these proteins causing nearby cells to increase their anti-viral defenses by activating natural killer cells and macrophages, up-regulating antigen presentation by increasing the expression of major histocompatibility complex (MHC) antigens and interfering (hence the name) with virus replication.  

Uniquely, many bats show constitutive expression of three interferon-alpha genes in the absence of viral nucleic acids, thereby having their anti-viral defense mechanism primed before infection and providing an environment that will tamp down viral load. Such constitutive expression of interferon would have dire consequences in other mammals since it would cause widespread inflammation suggesting that bats have anti-inflammatory mechanisms not possessed by other mammals and it is thought that these may have evolved to counteract cellular damage that might occur as a result of their being the only flying mammal.  

Because of the constant high interferon-alpha levels, bat-borne viruses may be shed at low levels from bat cells without eliciting a strong antibody response. Viral adaptation to these conditions in bats leads to other hosts (including humans) being more prone to develop pathology since they do not have the conditions for virus suppression.  The dampened inflammatory responses of bats to the presence of RNA viruses support the ideas that innate immune tolerance, rather than resistance, is the way that bats have evolved to host many virus species with limited disease.  

From DNA sequencing of bat genomes, it has been found that these animals lack a family of genes that encode proteins that are DNA sensors which recognize foreign DNA such as DNA viruses and damaged self DNA arising from RNA viral infection and the high metabolic rate of these animals. Perhaps this limits activation of the innate immune response leading to the avoidance of excessive inflammation. 

Despite recent advances in the understanding of bat viral tolerance, the consequences of this unique bat immunity on host-virus interactions and its implications for the understanding of new zoonotic disease have yet to be established and how bats limit excessive inflammation while asymptomatically hosting a greater variety of viruses is unknown.