Treasury Secretary Mnuchin announced this morning (March 20, 2020) that the filing date for 2019 individual tax returns has been moved to July 15, 2020. You will not incur penalty or interest if the return is filed and paid by then.
It appears that C corporations for calendar year 2019 are included in this announcement.
It is not clear whether S corporations and partnerships (which usually have a March 15 deadline for calendar year companies) will be granted the additional time. I have already filed extensions for my March 15th deadline clients; my guess (and this is my guess, not official policy) is that companies that did not file timely extensions by March 15 will be granted relief.
Payroll tax deposits and estimated tax payments do not appear to be included in this extension. Please continue to make timely payroll tax deposits, as penalties are severe and it takes considerable time and effort to get penalties waived. My guess (see above) is that some relief will occur.
If you are due a refund, file as soon as possible so you have access to the funds.