There is a lot of confusion in this thread about the distinction between emotional support animals (ESAs) and service animals, so let me clarify the legal framework as someone who has worked with tenants on these issues for years.
Under the Fair Housing Act (42 USC Section 3604), landlords and HOAs are required to make reasonable accommodations for tenants with disabilities, including allowing ESAs even when there is a no-pets policy. An ESA does not need specialized training -- it provides emotional support through companionship. However, you do need documentation from a licensed healthcare provider establishing that you have a disability-related need for the animal. This is not the same as buying a certificate online, which HUD has specifically warned against in its 2020 guidance (FHEO-2020-01).
The HUD guidance from January 2020 provides important clarification on what documentation landlords can and cannot require. They can ask for a letter from your healthcare provider (therapist, psychiatrist, doctor) confirming that you have a disability under the FHA and that the ESA is necessary to afford you equal opportunity to use and enjoy your dwelling. They cannot require specific diagnoses, detailed medical records, or information about the nature or severity of your disability.
Landlords can deny an ESA request only in narrow circumstances: if the specific animal poses a direct threat to health or safety that cannot be mitigated, or if the accommodation would impose an undue financial and administrative burden. Breed or size restrictions generally do not apply to ESAs, though there can be legitimate case-by-case safety concerns.
One critical point: online ESA letter mills (websites that charge 100-200 dollars for a letter from a provider you have never met) are increasingly being rejected by courts and HUD. The best practice is to get documentation from a provider who has an ongoing therapeutic relationship with you. A letter from a therapist you have been seeing for six months carries far more weight than one from an online service.
If your landlord or HOA refuses a legitimate ESA request, file a complaint with HUD or your state fair housing agency. The complaint process is free, and HUD takes these cases seriously. Damages can include actual damages, injunctive relief, attorney fees, and civil penalties.