A native of Suwanee, Georgia, Roby attended Peachtree Ridge High School, where he was an All-State receiver and defensive back for the Lions program.

On November 20, 2013, Ohio State head coach Urban Meyer announced that Roby would be leaving school early to enter the 2014 NFL draft.

During the draft process, Roby had private meetings and workouts with multiple teams, including the San Diego Chargers, Pittsburgh Steelers, Buffalo Bills, Detroit Lions, and Cincinnati Bengals.

[8][9][10] At the conclusion of the pre-draft process, Roby was projected to be a first or early second round pick by NFL draft experts and scouts.

In Week 6, Roby made three combined tackles, two pass deflections, and had his first career sack on New York Jets quarterback Geno Smith in the Broncos' 31–17 victory.

Roby finished his rookie season in 2014 with a career-high 65 combined tackles (63 solo), 13 pass deflections, two interceptions, one sack, and a forced fumble in 16 games and two starts.

On January 11, 2015, Roby appeared in his first career playoff game and recorded two solo tackles, a pass deflection, and intercepted Andrew Luck during a 24–13 loss to the Colts in the AFC Divisional round.

Newly appointed head coach Gary Kubiak named Roby the starting nickelback and the third cornerback on the team behind Aqib Talib and Chris Harris Jr to begin the regular season.

[25] In Week 2, Roby returned a fumble by Jamaal Charles for a touchdown in the final seconds of the fourth quarter to give the Broncos a 31–24 victory over the Kansas City Chiefs.

The following week, he made two solo tackles, deflected a pass, and intercepted Matthew Stafford during the Broncos' 24–12 victory at the Detroit Lions.

He finished the 2015 season with 40 combined tackles (34 solo), ten pass deflections, an interception, and a fumble recovery in 16 games and four starts.

The following week, he intercepted a tipped pass by teammate Aqib Talib on a two-point conversion to preserve a fourth-quarter 20–18 victory over the Patriots.

[28] On February 7, 2016, Roby appeared in Super Bowl 50 and recorded two solo tackles and three pass deflections in the Broncos' 24–10 victory over the Carolina Panthers.

[31] On October 30, 2016, Roby recorded a season-high seven combined tackles, deflected two passes, and returned an interception off Philip Rivers for a 49-yard touchdown during a 27–19 victory over the Chargers.

In Week 13, he collected three solo tackle, defended two passes, and returned an interception by Blake Bortles for a 51-yard touchdown in Denver's 20–10 win at the Jacksonville Jaguars.

[32] Without quarterback Peyton Manning leading the offense, the Broncos finished the 2016 season third in the AFC West with a 9–7 record and did not qualify for the playoffs.

[38] With Roby having impressed in his years as a backup, the general consensus heading into the 2018 season was that he was a high-quality defensive back deserving of a full-time starting role.

[39] This consensus convinced Broncos president of football operations John Elway to move on from Talib, trading him to the Rams, and to promote Roby to one of Denver's two starting cornerbacks.

[43] In week 13 against the New England Patriots on Sunday Night Football, Roby recorded his first interception of the season off a pass thrown by Tom Brady and returned it for 22 yards during the 28–22 win.

[44] In week 16 against the Tampa Bay Buccaneers, Roby intercepted a pass thrown by Jameis Winston and returned it for a 27-yard touchdown during the 23–20 win.

[46] In Week 6 against the Tennessee Titans, Roby recorded his first interception of the season off a pass thrown by Ryan Tannehill during the 42–36 overtime loss.

[58] On August 21, 2013, the case was conditionally dismissed after Roby pleaded guilty to misdemeanor disorderly conduct and agreed to complete a pre-trial diversion program.

[61][59] On April 29, 2014, Roby pleaded guilty to physical control of a motor vehicle while under the influence of drugs or alcohol and was charged with a suspended sentence of 180 days in jail.