OpenMW is a free, open source, and cross-platform recreation of the Gamebryo game engine used in Bethesda's The Elder Scrolls III: Morrowind open world role playing game. Not only is it fully compatible with original game files and mods, it also provides tools to create full standalone games. OpenMW also comes with numerous improvements over the original game engine including improved UI and mod support, improved AI and physics, and a work-in-progress Android port that supports touchscreen controls.
Daggerfall Unity, as the name suggests, is a complete recreation of Daggerfall game in unity game engine. It is free, open source, cross platform and comes with numerous enhancements like improved lighting, graphics, controls, mod support and full compatibility with modern systems. The original Daggerfall game is now available for free and you will need the original game files to play Daggerfall Unity. You can find more information about installation here.
Xash3D is a reimplementation of GoldSrc engine used in Half-Life 1. It can be used to play the original Half-Life game as well as numerous mods that are compatible with it. Xash3D was later forked into Xash3D FWGS bringing in improved multiplayer support, full Android compatibility and controller support. Android port download links are available here.
OpenTTD is a reimplementation of “Transport Tycoon Deluxe” strategy and simulation game. It comes with numerous enhancements, most notable of which are improved graphics, compatibility with modern systems, quality of life updates and improved multiplayer support. All the game assets have been recreated in OpenTTD, so you don't need to have the original game files (original files are still supported though). An Android port is available here.
GemRB is a free and open source recreation of the Infinity game engine developed by Bioware. The game engine was originally used in isometric role playing games like Baldur's Gate series, Icewind Dale series and Planescape: Torment. GemRB has brought numerous improvements over the original engine, including improved mod support, Android, and iOS ports. GemRB requires original games files to work, but a built-in demo is provided that can be used to test the engine and doesn't require any external files.
Note that remastered enhanced versions of Baldur's Gate series, Planescape: Torment and Icewind Dale 1 have been released by Beamdog in the recent past. These remastered versions have Android and iOS ports as well. If you have original games stored somewhere, you can use GemRB to play these games and skip purchasing the enhanced versions. If money is not an issue, I would recommend getting enhanced versions as they have a lot of quality of life improvements. Enhanced edition for Icewind Dale 2 doesn't exist as the original source code was lost, so the only option to play it on modern systems is to use GemRB.
ScummVM is a reimplementation of game engine used in classic point and adventure games made by game companies like LucasArts and Sierra On-Line. Some of these popular games include Broken Sword and Monkey Island titles. The “VM” part stands for “Virtual Machine” and in all it supports more than 250 games that can be played on Windows, macOS, Linux, Android, iOS, and even on consoles. ScummVM is free and open source and it has mainly brought in improved controls and compatibility with modern systems while preserving the original look and feel of the games. In fact, some of the original game makers have partnered with ScummVM to re-release their games on modern systems. Note that ScummVM is not an emulator but a complete recreation of the original engine and you will need original game files to play.
CorsixTH is a recreation of the popular hospital management and simulation game “Theme Hospital”, released back in 1997. It is free, open source, and cross-platform but you need original games files to play the games as it doesn't come with its own assets. An unofficial Android port is also available on the web. CorsixTH has mainly brought in high resolution support and compatibility with modern systems.
VCMI Project is a free and open source implementation of Heroes of Might and Magic III game engine. It comes with numerous improvements, a few of them include support for high resolution displays, improved GUI, improved game launcher and configuration tool, improved mod support, and compatibility with modern systems. Besides builds for Windows, macOS, and Linux, an Android port is also available. VCMI Project requires original game files to work.
OpenRA is a recreation of game engine used in classic real time strategy games like Command & Conquer series, Tiberian Dawn and Dune 2000. It comes with many improvements such as support for modern screen resolutions, improved controls and UI interface, improved hotkeys support, improved mods support and full built-in support for multiplayer games. The player base is quite active and you don't even need original game files to play. When first running OpenRA, you will be prompted to download game files.
RBDOOM-3-BFG is a source port of Doom 3 BFG first person shooter game originally released by id Software. The game uses id Tech 4 game engine whose source code was released in 2011. RBDOOM-3-BFG's main features include 64bit cross-platform builds, improved shadows, lighting and graphics, improved mods support, and compatibility with modern systems. To play DOOM 3 BFG using RBDOOM-3-BFG port, you will need original game files. An unofficial Android port of RBDOOM-3-BFG is available on the web.
EasyRPG is a free and open source reimplementation of RPG Maker 2000 and RPG Maker 2003 game engines. It can be used to play existing RPG Maker 2000 and 2003 games, provided you have the original game files. EasyRPG also comes with an editor that can be used to create games from scratch. EasyRPG Player can be used to play games on all major desktop operating systems, on some consoles, Android phones and iOS devices.
OpenRCT2 is a free and open source adaptation of the RollerCoaster Tycoon 2 game. Besides one to one reimplementation of the original game, it comes with many additional features like an improved editor, improved multiplayer gameplay, compatibility with modern systems including Android devices, improved UI, and so on. OpenRCT2 requires original game files to work, so you must own the RollerCoaster Tycoon 2 game.
Open source recreations of old games are not only allowing players to relive nostalgia, but they also help in preserving classics. Some of the projects mentioned above are in development for over a decade and they are community projects with no corporate backing.