This is a question which can have many answers, as the answer depends on what one considers an "epic." In the strictest sense of the word and according to the eNotes link below, an epic will have most or all of the following characteristics:
1. a setting remote in time and place
2. an objective, lofty, dignified style
3. a central incident or series of incidents dealing with legendary or traditional material
4. a theme involving universal human problems
5. a towering hero of great stature
6. superhuman strength of body, character, or mind
7. superhuman forces entering the action
Some ancient works (such as The Iliad, The Odyssey, and The Aeneid) certainly fulfill all these criteria; however, more modern works probably only meet several of them. (The "towering hero" and the supernatural elements are not as evident in modern literary works, though of course there are some notable exceptions, such as the Lord of the Rings trilogy.)
In general terms, a modern epic displays effective writing, contains universal themes, and has characters who are worthy of emulation despite their flaws. An epic must rise above the level of good and even great, and usually it is a work which has stood the test of time. For me it is the timeless books which still speak to the condition of man and his soul and have characters who are worthy of emulation despite their flaws that should be considered epics.
A list of modern American epics would probably include such novels as The Grapes of Wrath, To Kill a Mockingbird, Moby Dick, and The Scarlet Letter. A list of British epics is likely to include works of poetry, such as Paradise Lost, Beowulf, and The Divine Comedy, and Russian epics might be War and Peace and Crime and Punishment. Epic literary works from the rest of the world might include El Cid and Don Quixote and the play Cyrano De Bergerac. I have included a link to one list of epic literature, though it is certainly not exhaustive and is open to discussion.
Unfortunately, the word epic, like the word tragedy, has become rather prosaic and is used to refer to things which do not adhere to the more formal definition of the word.